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ESSENTIALS  OF  ALGEBRA 

COMPLETE   COURSE 

(AN  ADEQUATE  PREPARATION  FOR  THE  COLLEGE  OR  TECHNICAL  SCHOOL) 

FOR 

SECONDARY.  SCHOOLS 


BY 

JOHN   C.    STONE,    A.M., 

MICHIGAN  STATE  NORMAL  COLLEGE,   CO-AUTHOR  OF  THE 
SOUTHWORTH-STONE  ARITHMETICS 


JAMES   F.    MILLIS,    A.M., 

THE  SHORTRIDGE  HIGH  SCHOOL,   INDIANAPOLIS,   INDIANA 


OV  TTOAA'    ttA.Aa  TToXv 


BENJ.   H.    SANBBrN  &  CO. 

BOSTON  NEW  YORK  CHICAGO 


Wv^V  J^/^ 


S7^ 


Copyright,  1905, 
Br  JOHN  O.  STONE  and  JAMES  F.  MILLI8. 


U  '■ 


PREFACE. 

That  there  is  a  place  for  a  new  Algebra  recent  corre- 
spondence has  abundantly  proven.  This  book  contains  features 
which  will  not  only  arouse  and  sustain  interest  in  the  subject, 
but  are  now  demanded  in  an  elementary  course  in  algebra. 

It  is  believed  that  the  book  is  modern  and  progressive,  yet 
free  from  fads,  and  in  no  sense  extreme.  It  is  simple  in  style 
and  rigorous  in  treatment.  In  order  to  make  this  possible,  some 
of  the  topics  commonly  treated  in  elementary  algebra,  difficult 
for  the  beginner,  and  of  comparatively  little  value  to  him,  have 
been  postponed  or  omitted.  Highest  common  factors  by  divis- 
ion, and  square  and  cube  roots  of  polynomials  and  of  arith- 
metical numbers  by  formulse,  have  been  put  into  the  Appendix. 
These  may  be  read  whenever  the  teacher  thinks  advisable,  or 
they  may  be  omitted  without  detriment  to  the  subsequent 
work.  The  fundamental  laws  of  numbers  have  been  explained 
and  carefully  illustrated  when  introduced,  but  rigorous  proofs 
of  these  laws  have  been  put  into  the  Appendix,  where  they  may 
be  read  when  the  pupil  has  become  sufficiently  familiar  with 
algebraic  processes.  The  beginning  pupil  should  not  be  over- 
burdened with  the  proof  of  certain  simple  principles ;  yet  he 
must  see,  before  he  leaves  the  subject,  that  there  is  a  demon- 
stration for  every  principle. 

The  following  are  some  of  the  special  features  of  the  book. 

iii 

1  '4732 


iv  PREFACE 

Extension  of  number.  The  notion  of  a  general  number,  rep- 
resented by  some  letter  of  the  alphabet,  is  introduced  by  many 
illustrations.  The  pupil  is  shown  that  all  of  the  new  kinds  of 
numbers, — negative  numbers,  surds,  imaginary  numbers — like 
fractions,  arise  logicaMy  from  an  attempt  to  make  the  funda- 
mental processes  universal. 

Checks.  The  pupil  is  taught  from  the  beginning  how  to 
check  his  work — multiplication,  factoring,  etc. — by  substitut- 
ing particular  values  for  the  general  numbers  in  the  identities 
which  he  obtains.  This  gives  him  constant  practice  in  the  in- 
terpretation of  the  symbols  of  algebra,  broadens  his  grasp  of 
the  significance  of  general  number,  and  trains  him  to  be  ac- 
curate and  self-reliant. 

Factors.  The  subject  of  rational  factors  is  thoroughly  dis- 
cussed, and  made  the  basis  of  subsequent  work  whenever 
possible.  The  pupil  is  taught  to  reduce  all  expressions  to  be 
factored,  when  possible,  to  one  of  the  fundamental  type- 
forms  ;  otherwise  to  try  the  remainder  theorem.  Abundant 
miscellaneous  exercises  are  given. 

The  equation.  Special  care  has  been  taken  in  the  treatment 
of  equations.  The  pupil  is  required  to  see  that  every  step  in 
the  solution  of  an  equation  is  the  application  of  a  fundamental 
principle.  "  Transpose  "  and  "  Clear  of  fractions  "  are  terms 
}vhich  the  pupil  is  allowed  to  use  only  after  he  knows  the 
meanings  of  the  processes  which  they  represent. 

Correlation  with  science.  The  formula.  Any  letter  is  used 
as  an  unknown  number  and  formulae  are  solved  for  any 
letter  in  them.  For  tliis  purpose  many  fundamental  formulge 
are  borrowed  from  science.     Consequently,  the  algebra  which 


PREFACE  y 

the  pupil  will  meet  in  his  subsequent  science  work  will  be 
familiar  to  him. 

Expression  of  laws  by  equations.  It  is  shown  that  the  equa- 
tion may  be  used  to  express  a  law  of  science.  Practical  appli- 
cation is  thus  made  of  the  theory  of  variation.  Exercise  is 
given  in  interpreting  the  law  expressed  by  an  equation,  and  in 
writing  the  equation  which  expresses  a  stated  law.  The  sim- 
ple laws  and  formulae  of  science  are  used  in  the  work. 

The  graph.  The  graph  is  used  in  the  treatment  of  indeter- 
minate equations,  both  linear  and  quadratic,  as  a  means  of 
illustration.  It  is  used  to  illustrate  the  different  kinds  of  sys- 
tems of  equations,  such  as  impossible  systems,  etc.  The 
graphic  solution  of  a  system  is  shown,  and  the  solution  of  a 
single  equation  in  a  single  unknown  is  found  by  solving  a 
system. 

Exercises.  The  exercises  are  numerous,  new,  and  well 
graded.  They  are  sufficiently  difficult  to  call  for  effort  on  the 
part  of  the  pupil. 

Reviews.  Miscellaneous  review  exercises  are  distrilmted 
throughout  the  book.  These  reviews  consist  of  many  exer- 
cises and  questions  intended  to  bring  out  in  review  the  funda- 
mental principles  of  algebra. 

The  Brief  Gourse  consists  of  the  first  twenty-one  chapters 
of  the  Complete  Course  and  is  intended  for  classes  that  go  only 
through  quadratics,  inequalities,  ratio  and  proportion,  and  the 
theory  of  exponents.  The  Complete  Course  furnishes  an  ade- 
quate preparation  for  the  College  or  the  Technical  School. 

The  authors  take  pleasure  in  acknowledging  their  indebt- 
edness  for    many  valuable    suggestions  to   Professor  E.   A. 


Vi  PREFACE 

Lyman  and  to  Dr.  N.  A.  Harvey  of  the  Michigan  State 
Normal  College,  who  have  read  both  the  manuscript  and  the 
proof;  to  Professor  J.  L.  Love  of  Harvard  University,  and 
John  A.  Avery,  of  the  English  High  School,  Somerville,  Mass- 
achusetts, who  have  carefully  read  all  of  the  manuscript ;  and 
to  E.  Harry  English,  Supervisor  of  Mathematics  in  the  Wash- 
ington (D.  C.)  High  Schools,  who  has  carefully  read  the  first 
proof,  and  made  helpful  suggestions. 

J.  C.  S.  • 
June,  1905.  J.  F.  M. 


PUBLISHER'S  NOTE 

Before  accepting  the  manuscript  of  this  Algebra  it  was 
placed  in  the  hands  of  Professor  James  Lee  Love  of  the  Law- 
rence Scientific  School  of  Harvard  University,  who  gave  it  his" 
cordial  approval  and  endorsement. 

The  Southworth-Stone  Arithmetics,  published  in  January, 
1904,  have  in  one  year  been  introduced  in  more  schools,  ex- 
ceeded in  sale  any  competitors,  and  received  more  general 
commendation  from  the  leading  superintendents  and  teachers 
than  any  other  Arithmetics  ever  published  in  this  country. 
'Many  of  the  features  that  have  made  these  Arithmetics  so 
deservedly  popular  have  been  introduced  in  this  Algebra,  and 
the  book,  with  its  large  clear  type  and  open  page,  will  be  cor- 
dially welcomed  by  the  educational  public. 


CONTENTS. 


CHAPTER  I. 

PAGE 

Introduction 1 

Definitions 1 

Signs  of  Grouping " . .  6 

General  Number 8 

CHAPTER  II. 

Fundamental  Laws  of  Numbers 16 

Fundamental  Laws 16 

The  Equation 19 

Axioms 21 

CHAPTER  III. 

Negative  Number 30 

Definitions 30 

Addition  of  Algebraic  Numbers 35 

Subtraction  of  Algebraic  Numbers 37 

Multiplication  of  Algebraic  Numbers 39 

Division  of  Algebraic  Numbers 41 

Exercises  for  Review  I 43 

CHAPTER  IV. 

Addition  and  Subtraction  of  Literal  Expressions 46 

Addition  of  Monomials 46 

Addition  of  Polynomials 47 

Subtraction 49 

Removing  Signs  of  Grouping , .  53 

Inserting  Signs  of  Grouping 58 

vii 


Viii  CONTENTS 

CHAPTER  V. 

PAGE 

Multiplication  of  Literal  Expressions 54 

Laws  of  Exponents 54 

Multiplication  of  Monomials 54 

Multiplication  of  Polynomials  by  Monomials 56 

Multiplication  of  one  Polynomial  by  Another 60 

CHAPTER  VI. 

Division  of  Literal  Expressions 64 

Law  of  Exponents 64 

Meaning  of  a", ....  64 

Division  of  Monomials 64 

Division  of  a  Poljaiomial  by  a  Monomial 65 

Division  of  one  Polynomial  by  Another 66 

The  Fraction 71 

CHAPTER  VII. 

Powers  and  Roots 75 

Powers 75 

Roots  :.) 83 

Roots  of  Monomials 86 

Roots  of  Polynomials  by  Inspection 90 

CHAPTER  VIII. 

Special  Products  and  Quotients 92 

Products 92 

Quotients 96 

Exercises  for  Review  II 100 


CHAPTER  IX. 

Factors 103 

Factors— Type  Forms 103 

The  Remainder  Theorem 123 

Synthetic  Division 125 


CONTENTS  ix 

CHAPTER  X. 

PAGE 

Common  Factors  and  MuLTiPiiES 130 

Definitions 130 

Highest  Common  Factor 131 

Lowest  Common  Multiple 135 

CHAPTER  XI. 

Fractions   139 

Definitions,  Laws,  etc 139 

Addition  and  Subtraction  of  Fractions 146 

Multiplication  of  Fractions  150 

Division  of  Fractions , 153 

Complex  Fractions 155 

Exercises  for  Review  III 157 

CHAPTER  XII. 

Linear  Equations — One  Unknown 161 

Definitions , 161 

The  Solution  of  a  Linear  Equation 165 

Fractional  Equations 167 

The  Formula 170 


CHAPTER  XIII. 

Linear  Equations— Systems 180 

Definitions 180 

Elimination 183 

Systems  of  Fractional  Equations 190 

The  Graph  of  an  Equation 193 

Graph  of  a  System 197 

Exercises  for  Review  IV 209 

CHAPTER  XIV. 

Surds  and  Imaginary  Numbers 211 

Definitions,  Reductions,  etc 211 

Addition  and  Subtraction  of  Surds 215 


X  CONTENTS 

PAGH 

Rationalization ^ ,  220 

Powers  and  Roots  of  Surds 221 

Imaginary  Numbers. ... 223 

Geometric  Representation  of  Imaginary  Numbers 228 

CHAPTER  XV. 

Quadratic  Equations— One  Unknown 230 

Definitions 230 

Solution  of  Pure  Quadratics 231 

Solution  of  Complete  Quadratics 233 

The  Discriminant 241 

Solving  a  Quadratic  Formula 244 

CHAPTER  XVI. 

Higher  Equations— Equations  involving  Surds— One  Unknown  253 

Solution  of  Equations  of  Higher  Degree 253 

Solution  of  Equations  involving  Surds 256 

Introduction  of  New  Solutions 257 

CHAPTER  XVII. 

Systems  involving  Quadratic  and  Higher  Equations 261 

Solution  of  Systems 261 

Special  Devices 270 

Graphs  of  Systems  of  Quadratics 273 

Exercises  for  Review  V 285 

CHAPTER  XVIII. 

Inequalities 291 

Definitions 291 

Principles 292 

CHAPTER  XIX. 

Ratio  and  Proportion 298 

Definitions 298 

Principles , 300 


CONTENTS  Xi 

CHAPTER  XX. 

PAGK 

Variation.    Algebraic  Expression  of  Law 310 

Definitions,  etc 310 

Law  expressed  by  an  Equation  314 

Limits  of  Variables '. 318 

Properties  of  Zero 321 

~  CHAPTER  XXL 

Fractional  and  Negative  Exponents 324 

Definitions  and  Interpretations 324 

Exercises  for  Review  VI 334 

CHAPTER  XXII. 

Permutations  and  Combinations 337 

Permutations, — Definitions,  etc 337 

Combinations, — Definitions,  etc 342 

CHAPTER  XXIII. 

The  Binomial  Theorem 346 

Proof — The  Exponent  a  Positive  Integer 346 

Exponent  Negative  or  Fractional 350 

Extraction  of  Roots  by  use  of  Binomial  Theorem 352 

CHAPTER  XXIV. 

Progressions. 354 

Series, — Definitions,  etc 354. 

Arithmetical  Progression 355 

Geometrical  Progression ". 362 

Harmonical  Progression 369 

Exercises  for  Review  VII 371 

CHAPTER  XXV. 

Undetermined  Coefficients 375 

Tlieorem  of  Undetermined  Coefficients 376 


Xii  CONTENTS 

PAGE 

Expansion  of  Fractions  into  Series 378 

Expansion  of  Surds  into  Series ... 380 

Reversion  of  Series  382 

Partial  Fractions 383 

CHAPTER  XXVI. 

Logarithms 388 

Exponential  Equations, — Definitions,  etc 388 

Fundamental  Principles  of  Logarithms 390 

Exponential  Equations 406 

Table  of  Mantissas t 411 

Exercises  for  Review  VIII 407 


APPENDIX. 

Square  and  Cube  root  by  the  formula 1 

The  Highest  Common  Factor  and  Lowest  Common  Multiple.  16 

The  proof  of  the  Fundamental  Laws : 23 


ESSENTIALS  OF  ALGEBRA 


CHAPTER  I. 
INTRODUCTION. 


1.  Algebra.  No  clear  line  of  distinction  will  be  found 
between  algebra  and  arithmetic.  Algebra  deals  with  number 
and  with  the  same  fundamental  processes  which  are  dealt  with 
in  arithmetic.  The  processes  of  arithmetic,  however,  are 
extended  in  algebra ;  and  the  meaning  which  is  attached  to 
number  in  arithmetic  is  also  extended  in  algebra.  Algebra 
deals  particularly  with  the  equation. 

2.  Signs.  The  signs  +,  — ,  X,  -^,  and  =,  which  are  used  in 
arithmetic,  are  used  with  the  same  meanings  in  algebra. 
Other  signs  used  in  algebra  will  be  introduced  later,  when 
they  are  needed. 

3.  Fundamental  processes.  The  fundamental  processes  of 
algebra,  as  in  arithmetic,  are  addition,  subtraction,  multipli- 
cation, and  division. 

a.  Addition.  The  addition  of  two  or  more  numbers  is  the 
process  of  uniting  them  into  oiie  whole. 

As  in  arithmetic,  two  numbers  to  be  added  are  called 
addends,  and  the  result  is  called  the  sum. 

To  indicate  addition  the  sign  +  is  placed  between  the  two 
numbers  to  be  added. 

Thus,  7  +  3  indicates  that  3  is  to  be  added  to  7,  The  result, 
which  is  10,  is  called  the  sum. 


2  ALGEBRA 

The  symbol  =,  placed  between  two  numbers,  indicates  that 
they  have  the  same  value.  Thus,  7  +  3=10.  This  expression  is 
called  an  equation,  and  is  read  "seven  plus  three  equals  ten." 
We  shall  have  more  to  do  with  equations  later. 

Three  or  more  numbers  may  be  added  by  adding  two  at  a 
time. 

Thus,  to  add  3,  6,  5,  and  2;  adding  6  to  3  gives  9;  adding  5  to 
9  gives  14;  adding  2  to  14  gives  16,  the  sum. 

From  the  foregoing  it  is  evident  that  to  find  the  sum  in  a 
series  of  additions  we  may  proceed  from  left  to  right,  uniting 
two  numbers  at  a  time. 

Thus  to  find  the  sum  of  4  +  6  +  2  +  9^;  we  have  4  +  6=10; 
10*f2=:12;  12  +  9|=21^. 

b.  Subtraction.  Subtraction  is  the  inverse  of  addition.  Given 
one  of  two  numbers  and  their  sum,  subtraction  is  the  process 
of  finding  the  other  number. 

Subtraction  is  expressed  by  placing  the  sign  —  between 
the  two  given  numbers.  When^  so  used,  it  means  that  the 
second  number  is  to  be  subtracted  from  the  first. 

Thus,  9  —  5  indicates  that  5  is  to  be  subtracted  from  9. 

As  in  arithmetic,  the  number  obtained  by  subtracting  one 
number  from  another  is  called  the  difference,  or  remainder  ; 
the  number  to  be  subtracted  is  called  the  subtrahend  ;  and 
the  given  sum,  or  the  number  from  which  the  subtrahend 
is  to  be  subtracted,  is  called  the  minuend. 

From  the  definition  of  subtraction,  it  follows  that  to  sub- 
tract the  second  of  two  numbers  from  the  first  is  to  find  a 
third  number  such  that  if  it  be  added  to  the  second,  the 
sum  will  equal  the  first.  Hence  the  following  important 
principle : 

subtrahend  +  remainder  =  minuend, 


INTRODUCTION  3 

A  series  of  two  or  more  subtractions  may  be  performed 
by  proceeding  from  left  to  right,  subtracting  one  number  at 
a  time. 

Thus,  to  find  the  value  of  26  -  12^  -  3|  ;  26  -  12|  =  13^  ;  13^ 
— 31  =  10,  the  value. 

Likewise,  a  series  of  additions  and  subtractions  may  be 
performed  by  proceeding  from  left  to  right,  performing  one 
operation  at  a  time. 

Thus,  to  find  the  value  of  100  -  8  +  2|  +  10  -  25  ;  we  have  100-8 
=  92  ;  92  +  2|  =  94|  ;  941  +  10  =  104^  ;  104^— 25=79|. 

c.  Multiplication.  Multiplication  has  been  defined  in  arith- 
metic as  the  process  of  taking  one  number,  called  the 
multiplicand,  as  many  times  as  there  are  units  in  the  other, 
called  the  multiplier.  It  is  evident  that  this  definition  holds 
only  when  the  multiplier  is  a  whole  number,  and  fails  when  it 
is  a  fraction. 

Thus,  to  multiply  7  by  2|  would  mean  to  take  7  as  many 
times  as  there  are  units  in  2|,  that  is,  2|  times.  This  is  im- 
possible.    One  cannot  do  a  thing  2|  times. 

Instead  of  this  definition,  which  is  not  sufficiently  general, 
we  shall  use  the  following : 

To  rmdtiply  one  number^  called  the  midtiplicand^  by  another^ 
called  the  nudtiplier^  is  to  use  the  multiplicand  as  we  tnust  use 
unity  to  obtain  the  multiplier.      The  result  is  called  the  product. 

For  example,   let  us  multiply  3  by  4.     To  obtain  4  we  take 

1  +  1  +  1  +  1=4. 
Hence,  to  obtain  the  product,  we  take 
3  +  3  +  3  +  3=12. 
Here  we  have  done  to  3  what  we  did  to  1  to  obtain  4. 


^-^2 


GEBRA 
Again,  to  multiply  2  by  4|  ;  we  take 

Hence  the  product  i^jf^J^'J^^'^'  ^ y^ ^-^^  ::-  /J'/i^  _ 

2  +  2  +  2  +  2  +  I  +  |=8  +  ^=9|.  ^'^ 

This  definition  evidently  holds  when  the  multiplier  is 
either  an  integer  or  a  fraction,  hence  we  see  that  it  is  iiion 
geoieral  than  the  old  one. 

The  sign  commonly  used  in  arithmetic  to  indicate  mul- 
tiplication is  the  oblique  cross  X?  placed  between  the  num- 
bers. When  an  integral  multiplier  is  written  first,  the  sign 
is  read  '•  times  ; "  and  when  the  second  number  is  consi«  vired 
the  multiplier,  the  sign  is  read  "  multiplied  by." 

Thus,  8x7=56,  is  read  "  8  times  7  equals  56"  or  "  8  multi- 
plied by  7  equals  56."  In  this  book  the  second  number  will  be 
considered  the  multiplier,  thus,  5x2|  is  read  "5  multiplied 
by  2|." 

In  algebra  the  dot  (  ; )  also  indicates  multiplication  when 
placed  between  two  numbers,  in  a  position  above  that  which 
would  be  occupied  by  the  decimal  point. 

Thus,  8x7  may  be  written  8  •  7. 

A  series  of  two  or  more  multiplications  may  be  performed  by 
proceeding  from  left  to  right,  performing  one  multiplication  at 
a  time. 

Thus,  to  find  the  value  of  7  x  8  x  |  x  2  ;  we  have  7x8=56  ; 
56  x  1=  84  ;  84  X  2  =  168. 

The  result  obtained  by  a  series  of  two  or  more  multiplica- 
tions is  called  the  product  of  all  of  the  given  numbers.  In  the 
above  example  168  is  the  product  of  7,  8,  |  and  2. 

d.  Division.  Division  is  the  inverse  of  multiplication. 
Given  one  of  two  numbers -and  their  product,  division  is  the 
process  of  finding  the  other  number.     The  given  product,  or 


INTRODUCTION  5 

number  to  be  divided,  is  called  the  dividend ;  the  given  number, 
or  number  by  which  the  dividend  is  divided,  is  called  the 
divisor ;  and  the  result,  or  number  found,  is  called  the  quotient. 

Division  is  indicated  by  placing  the  sign  -=-  between  the  two 
given  numbers.  When  so  used,  the  second  number  is  the 
divisor. 

Thus,  for  "divide  10  by  2,"  we  write  10-^2. 

In  algebra  division  is  often  indicated  by  use  of  the  horizon- 
tal line  — ,  oblique  line  /,  or  colon  : ,  placed  between  the  dividend 
and  divisor. 

Thus,  10^2  may  be  written  >/,  10/2,  or  10  :  2  ;  read  "  10  divided 
by  2." 

From  the  definition  of  division  we  have  the  following  impor- 
tant principle : 

quotient  x  divisor = dividend. 

Thus,  since  20^4=5,  then  5x4=20. 

A  series  of  two  or  more  divisions  may  be  performed  by 
proceeding  from  left  to  right,  performing  one  division  at  a 
time. 

To  find  the  value  of  256^8^16  ;  we  have  256^8=32  ;  32^16=2. 

Likewise,  a  series  of  multiplications  and  divisions  may  be 
performed  by  proceeding  from  left  to  right,  performing  one 
operation  at  a  time. 

To  obtain  28-^4x^x2^3;  we  have  28-=-4=7  ;  7x9=63;  63x2 
=126  ;  126^3=42. 

4.  Expressions.  Any  combination  of  number-symbols  indica- 
ting one  or  more  processes  such  as  addition,  multiplication,  etc., 
is  called  an  expression. 

Thus,  2x6  —  3  +  1/5  is  an  expression. 


6  ALGEBRA 

A  rational  expression  is  an  expression  that  contains  only  indi- 
cated additions,  subtractions,  multiplications,  divisions,  and 
powers  of  numbers.     See  §13. 

5 .  Signs  of  grouping.  An  expression  is  often  used  as  a  si7igle 
number.  To  indicate  that  it  is  to  be  so  used  the  expression  is 
usually  enclosed  within  a  sign  of  grouping,  or  sign  of  aggrega- 
tion. 

We  shall  use  four  different  signs  of  grouping  to  enclose  ex- 
pressions.    They  are  the  parentheses  ( ),  the  brackets  [  ],  the 

braces    {  },  and  the  vinculum  ,   the    last    being    placed 

above  the  expression.  They  all  have  the  same  meaning,  and 
the  different  signs  are  used  in  different  places  merely  for  con- 
venience. They  indicate  that  the  expressions  enclosed  by  them 
are  to  be  used  collectively,  i.e.,  used  as  one  number. 

Thus,  9  —  (7—2)  indicates  that  7—2,  as  one  number,  is  to  be 
subtracted  from  9.  We  have  7—2=5  ;  9  — 5=4.  The  student  will 
note  carefully  the  difference  between  9  — (7— 2)  and  9—7—2. 

The  expression  9-(7-2)=9-[7-2]=9- {7-2;  =9-Tr2. 

Again,  6  x  (5  +  7)  indicates  that  6  is  to  be  multiplied  by  5  +  7  as 
one  number.     Thus  5  +  7=12  ;  and  6  x  12=72. 
24 4 

In  the  expression  ^ — 5—  the  horizontal  line  serves  three  pur- 

poses  at  once.  It  is  a  vinculum  under  the  24—4,  a  vinculum  over 
the  2  +  8,  and  a  sign  of  division.     Thus    ^:ii=?^  =2. 

It  follows  from  the  foregoing  that  in  expressions  containing 
signs  of  grouping.,  the  operations  indicated  within  the  signs  of 
grouping  must  be  perf or n^ed first. 

6.  Evaluation  of  expressions.  The  value  of  an  expression 
is  the  result  obtained  by  performing  all  of  the  operations  in- 
dicated within  it. 

Tlius,  the  value  of  38-4  +  12-3  -^  (2  +  3  +  4)-  ^^  is  34. 


INTRODUCTION  7 

MathematiciaiTs  have  agreed  that  in  expressions  containing 
additions,  subtractions,  multiplications,  and  divisions,  the 
multiplications  and  divisions  shall  take  precedence  over  the 
additions  and  subtractions.*  Hence  the  following  rule  is  to  be 
observed  in  the  evaluation  of  all  expressions : 

If  the  expression  contains  no  signs  of  grouping :  (1)  all 
operations  of  multiplication  and  division  should  be perfor7ned  in 
the  order  in  lohich  they  are  written  from  left  to  rights  before 
any  of  those  of  addition  and  subtraction  are  performed  ;  (2)  in 
the  resulting  expression  which  vnll  contain  only  additions  and 
subtractions^  these  should  then  be  performed  in  order  from  left 
to  right. 

If  the  expression  contains  signs  of  grouping^  all  operations 
indicated  within  these  must  'first  be  performed  according  to 
the  foregoing  rule.  -    \f^  >2^^     ^  ^  ^ 

Example  1.     Evaluate  98-(10-2  +  7)  +  (12-2)  -^  (1  +  4). 

Performing  the  operations  witfiin  the  parentheses,  we  have 
10—2  +  7=15;  12—2=10;  1  +  4=5.  Hence  the  expression  becomes 
98—15  +  10^5.  Performing  the  division,  we  have  98  —  15  +  2. 
Then    performing    the    subtraction    and    addition,     we    have 

98-15  +  2=85. 

Example  2.     Evaluate  ?-±i_-i^  +  7(2  +  3-1). 

We  have  ^_^  +  7  (2  +  3-1)  =-|-4 +7^4 

=3-2  +  28 
=29. 

EXERCISE    1. 

1.  Find,    by  using  the    definition  of    multiplication,   the 
product  7x5- 

2.  Find,  in  the  same  way,  the  product  5  X  ^^' 

*  The  reason  for  this  will  appear  later.  Members  connected  by  X 
or  -r  form  a  single  term.     §  15. 


8  ALGEBRA 

3.  If  the  subtrahend  is  28  and  the  remainder  15,  what  is 
the  minuend  ? 

4.  If  the  divisor  is  8  and  the  quotient  12,  what  is  the 
dividend  ? 

5.  In  the  product  9x7  what  is  the  multiplier  ? 

6.  What  is  the  difference  between  8x5  and  5x8? 

7.  What  conclusion  may  you  draw  from  Problem  6  as  to 
the  relation  between  the  multiplicand  and  multiplier  in  any 
problem  ? 

Evaluate  the  following  expressions  : 

8.  12-7-3  +  6.  17.  (6-2)X(8  +  3). 

9.  16  +  4-3-10  +  8.         ■        18.  (5-3)- (5  +  3).  (8-7). 

10.  (16-4)-(8  +  2).  19.  (9  +  l)-(3  +  2)X(ll-8). 

11.  18  +  3  X  (7-2).  ^Q    7  +  19 

12.  6-(3  +  9)  ^(6-4).  '     '^-^' 

13.  4x(7  +  8-3).  21.    ^5^X3. 

14.  5x(3-2)  ^  (6  +  4).  ^^     2  +  12^8-(10-4)     3    ^^ 


15.  8  +  7x9-8-2x7-5.         "'""        7       '         12 

16.  3X7X2---6--7X4.         .     23.   jl5  +  3]-|15-3 

24.   2  +  3^.  (6-4)--(l  +  5) 


2      .-         ^    2. 


1  +  2  -^^.J 

26.  (6  +  5)  •  (8 -6) -^7^=^.        4 

26.  10-4--[13  +  2-7]    15--9r=^ 

GENERAL  NUMBER. 

7.  In  the  following  paragraphs  we  shall  discuss  a  new  kind 
of  number. 

The  student  understands  that  the  character  or  figure  3  is 
not  a  number,  but  that  it  is  a  symbol  which  represents  a  num- 
ber. All  of  the  numerals  2,  7,  102,  |,  5J,  etc.,  and  the  Roman 
characters  I,  V,  X,  etc.,  represent  numbers  with   which  the 


INTRODUCTION  9 

student  is  familiar.     We  shall  see  that  there  are' other  symbols 
which  may  represent  number. 

8.  Definite  number.  Each  of  the  symbols  1,  2,  3,  4,  etc., 
represents  a  number  with  one  definite  value.  For  example,  3 
represents  the  single  number-idea  which  we  have  learned  to 
call  three.  The  numbers  which  these  symbols  represent  have 
the  same  values  in  all  problems  in  which  they  may  occur. 
Hence  they  are  called  definite,  particular,  or  single-valued 
numbers. 

9.  General  number.  On  the  other  hand,  it  is  sometimes  con- 
venient to  use  number  symbols  that  can  represent  more  than 
one  value.  Such  a  number  evidently  can  not  be  represented 
by  one  of  the  symbols  1,  2,  3,  4,  5,  etc.  It  is  usually  repre- 
sented by  some  letter  of  the  alphabet.,  as  a  or  x.  Note  carefully 
the  following  illustrations. 

(1)  To  get  the  cost  of  a  number  of  similar  articles  at  a  certain 
price  per  article,  we  always  multiply  the  price  of  one  article  by 
the  number  of  articles.     In  other  words  we  always  use  the  rule  : 

cost  of  all  articles ={pribe  of  one  article)  x  {number  of  articles). 

Thus,  to  get  the  cost  of  7  books  at  $1.25  per  book,  we  have 
cost  of  7  books  =$1.25x7 

=$8.75. 

A-lso,  to  get  the  cost  of  12  pencils  at  5  cents  a  pencil,  we  have 
cost  of  12  pencils=5c  x  12 

=60  cents. 

Now,  if  in  place  of  the  statement  ^'■cost  of  all  articles,^'  which 
represents  a  number,  we  put  c,  the  first  letter  of  "cost"  ;  in 
place  of  ''price  of  one  article,'' we  put  p,  the  first  letter  of 
''  price"  ;  and  in  place  of  ''number  of  articles,''  we  put  n,  the 
first  letter  in  "  number  "  ;  the  above  rule  becomes  c=p  x  n. 

This  rule,  c=pxn,  in  which  c,  p,  and  n  represent  immbers,  ap- 


10  ALGEBRA 

plies  to  all  particular  problems  of  the  above  sort.  The  letters  c, 
p,  and  w,  represent  different  particular  values  in  different  partic- 
ular problems.  Thus,  in  the  book  problem  above,  c=$8.75,  p= 
$1.25,  n=7  ;  in  the  pencil  problem,  c=60c,  p=:5c,  and  n=12. 

(2)  Simple  interest  on  money  is  always  obtained  by  the  rule 

interest  =  principal  x  rate  x  time. 

Thus,  $200  at  6^,  for  3  years,  would  give 
interest  =  $200  x.  06x3 
=$36. 
Now,  if  in  the  rule 

interest  =  principal  x  rate  x  time, 

we  replace  each  word  by  its  first  letter,  we  get 

i  =  p  X  r  X  t^ 

in  which  i,  p,  r,  and  f,  represent  numbers.  These  letters  stand 
for  different  particular  numbers  in  different  problems. 

Note.     It  is  understood,  of  course,  that  "  time  "  refers  to  the  ab- 
stract number  representing  the  number  of  years. 

(3)  The  distance  an  object,  as  a  train,  has  moved  in  a  given 
time  is  found  by  the  following  rule  : 

distance  =  rate  x  time. 
In  the  same  manner  as  above,  this  may  be  written 
d  =  r  X  t. 
d,   r,   and  t  represent  different  values  in    different  particular 
problems. 

7  2        7  +  2 

(4)  The  statement      Trr+-=r()  ~~iT)"   i^  ^  P^^'ti^^^^^  ^^^^  of  the 

following  principle  in  addition  of  fractions  : 

first  numerator  second  numerator  _  first  num.  -\-  Sftcmid  num. 
common  denom.      common  denom.      ~  common  denom. 

This  may  also  be  written 
which  expresses  m  a  way  the  rule  for  adding  fractions.     Here, 


c         c  c 


INTRODUCTION  H 

/,  s,  c,  represent  numbers  which,  in  one  set  of  fractions,  will 
have  certain  particular  values,  and  in  another  set  of  fractions, 
will  have  other  particular  values.    . 

It  must  now  be  evident  that  numbers  may  sometimes  be 
represented  by  letters  of  the  alphabet ;  *  and  that  a  number 
represented  by  a  letter  may  have  any  particular  value  v^hatever. 

Thus,  a  may  equal  1,  2,  3,  4,  |,  ||,  or  ayiy  other  definite  or  par- 
ticular number. 

Hence,  numbers  represented  \>j  letters  are  called  many- 
valued,  or  general  numbers.  General  numbers  are  largely  em- 
ployed in  the  problems  of  algebra. 

10.  The  same  laws  which  apply  to  particular  or  definite 
numbers  must  evidently  apply  also  to  general  numbers.  Gen- 
eral 7iumbers  may  be  added,  multiplied,  or  subjected  to  any 
other  operation  which  may  be  performed  upon  definite 
numbers. 

For  example,  a  +  6,  a  —  6,  a  x  b^  a-i-b,  represent  respectively 
the  sum,  difference,  product,  and  quotient  of  the  general  num- 
bers represented  by  a  and  b.  When  a  =  8  and  6  =  4,  what  are  the 
values  of  a  +  6,  a  X  6,  and  a-^b'i 

Hereafter,  instead  of  speaking  of  the  "  numbers  represented 
by  a,  5,  etc.,"  we  shall  speak  of  "  the  numbers  a,  ^,  etc." 

11.  Some  special  notation  in  the  nmltiplication  of  general 
numbers  may  now  be  considered. 

In  addition  to  the  use  of  the  two  signs  of  multiplication  dis- 
cussed in  §  3,  multiplication  may  sometimes  be  indicated  by 
the  absence  of  any  sic/n  between  the  given  numbers. 

*  By  attaching  superscripts  and  subscripts  to  the  letters  of  our  al- 
phabet, other  symbols  for  representing  numbers  may  be  obtained. 
Thus  a',  a",  a'",  etc.,  and  oci,  a2,  cca,  tti,  etc.  The  letters  of  the  Greek 
alphabet,  a,  /3,  y,  S,  etc.,  are  also  much  used  for  representing  numbers. 


12  ALGEBRA 

Thus,  a  X  6,  ab,  and  a&,  all  indicate  the  product  of  a  and  b. 
The  product  of  4,  a,  and  x  may  be  written  4ax. 
The  product  of  a  +  £c  and  b  +  y  may  be  written  (a+a?)  (b  +  y)^  as 
well  as  (a  +  a?)  x  {b-\-y),  or  (a+a^)-(6  +  2/). 

It  is  clear  that  on  account  of  the  place  value  feature  of  our 
notation  for  definite  number  by  which  a  figure  represents  both 
an  intrinsic  value  and  a  local  value,,  the  absence  of  a  sign  may 
not  be  used  to  indicate  the  product  of  two  definite  numbers. 
Thus  34  does  not  mean  3x4,  the  value  of  which  is  12,  but 
means  3x10  +  4. 

12.  Factors.  Coefficients.  Numbers  which  are  multiplied  so 
as  to  form  a  product  are  called  factors  of  the  product. 

Thus,  in  5aa?,  5,  a,  and  x  are  called  the  factors  of  ^ax. 

Any  factor  of  a  product,  or  the  product  of  any  two  or  more 
factors,  is  called  the  coefficient  of  the  product  of  the  remaining 
factors. 

For  example,  in  5aa?,  5  is  the  coefficient  of  ax,  5a  of  x,  a  of  5a?, 
etc. 

If  a  coefficient  is  a  definite  number,  it  is  called  a  numerical 
coefficient. 

For  example,  in  5aa?,  5  is  the  numerical  coefficient. 

If  no  numerical  coefficient  is  written,  1  is  understood. 

Thus,  abc  is  the  same  as  la6c. 

13.  Powers.  If  all  of  the  factors  of  a  product  are  equal,  the 
jjroduct  is  called  a  power  of  one  of  the  factors.  It  is  usually 
written  in  an  abbreviated  form. 

Thus,  aaaaaa  is  written  a^,  and  is  called  a  power  of  a. 
Similarly  xxxx  is  written  x*. 

In  the  power  a^,  a  is  called  the  base  and  6  the  exponent.  The 
exponent  is  a  number  written  at  the  right  of,  and  above  the 


INTRODUCTION  13 

base,  to  show  the  number  of  times  the  base  is  to  be  used  as  a 
factor  to  form  tlie  power. 

Thus,  a2,  read  "  a  square,"  or  "  a  second  power,"  denotes  aa;  a^, 
read  "  a  cube,"  or  "a  third  power,"  denotes,  aaa;  a*,  read  "a 
fourth  power,"  denotes  aaaa;  a^,  read  "a  fifth  power,"  denotes 
aaaaa  ;  a"\  read  "a  exponent  ?i2."  or  "a  mth  power",  denotes 
aaa  ...  to  the  product  of  m  factors.- 

If  no  exponent  is  written,  tlie^rs^  power  is  understood. 
Thus,  a  is  the  same  as  a^. 

14.  Literal  expressions.     An  expression  containing    one    or 
more  general  numbers  is  called  a  general  or  literal  expression. 
For  example,  2a  +  bG—x  is  a  literal  expi'ession. 

A  literal  expression  may  liave  any  definite,  or  particular 
value,  which  depends  upon  the  values  of  the  general  numbers 
involved  in  it.  A  definite  value  of  a  literal  expression  may  be 
found  wlien  a  definite  value  is  assigned  to  eacli  general  number 
involved  in  it. 

For  example,  when 

a  ='10  and  a;  =  6,  2a— x  =  14  ;  2ax  =  120  ;  3a^x  =  5. 

To  find  the  value  of  a  literal  expression  use  tlie  method  of 
§  6.     Note  carefully  the  following  examples. 

Example  1.  Find  the  value  of  a  +  d—c  +  b,  when  a=l,  6=2, 
c=3,  d=4. 

The  expression  becomes  1  +  4—3  +  2. 
Performing  additions  and  subtraction,  1  +  4  —  3  +  2=4. 

Hence  a+d— c  +  6=4. 

Example  2.  Find  the  value  of  3a^  6^  when  a=4,  6=2. 

We  have        3a^  ¥  =^aa666=3-4-4-2-2-2=384. 

Example  3.  Find  the  value  of  a  +  3S— 2^  +  (i-f-2,  when  a=l, 
6=2,  c=3,  d=4. 


14  ALGEBRA 

We  have  a  +  36—2c  +  <i^2=l  +  3x2-2x  3  +  4^2 
'       i  =1+6-6+2 

=  3. 

Example  4.  Find  the  value  of  a^—}fi  +  (3(i— a)  -^  (6  +  3c},  when 
a=2,  &=1,  c=3,  d=4. 

We  have  a^-h^  =aaa-bbb=2-2-2  -111=:8-1=7  ;  3d-a= 
3-4-2=12—2=10;  6  +  3c=l  +  3-3=l  +  9=10.  Hence  the  expres- 
sion becomes  7  +  10-i-10=7  +  l=8. 

h       15 
Example  5.  Find  the  value  of  a  +  ^y-^ — ^r-j-,  when  a=4,    6=24, 

c=3,  d=l. 

w^   u  ^         24         24        ^      15        15         15       ^ 

We  have  ^^=^  =  —  =  4;  -3^=^,  =-^=5. 

Hence  a+  ^ i^  =4  +  4-5=3. 

2c       Sd 


EXERCISE  2. 

If  a— 3,  5=6,  c  =  2,  c7=5,  a:;  =  4,  y  =  3,  find  the  value  of 

1.  h-d+a-c  +  y.  11.  {"Id-b)  {2b-d). 

2.  a  +  5c— a;.  12.   {x  +  yy. 

3.  5a^5.  13.  {a  +  b  +  ci'd)  +  {xfi/)., 

4.  5^-a^  j^      1       2       1 


5.  cd^ab^xy.  ^      ^ 

6.  a&x—^ac. 

7.  ba'b-^ed. 


c 


6.  ac^x-2ac.  15     ^  +  ^       ^  +  ^ 


^ — :-  b  +  x        b-\-x  , 

9.  .(.-f-3c-y).  ^^-^I7,ib-a)id-c),-'f. 

10.  b^cxy~a.  18.  («/=  +5^  +c^  -c^T- 

19.  a;^"  =  what  power  of  £c  ?     What  does  it  mean  ? " 

20.  Give  the  numerical  coefficients  in  4a-'5y  a^bb;  b-c. 

21.  In  7x\'  what   is  the   numerical  coefficient?    The   base? 
The  exponent? 


INTRODUCTION  15 

1 5 .  A  term  is  an  expression  in  which  the  numbers  are  not 
connected  by  the  sign  +  or  —  .  The  parts  of  an  expression 
which  are  connected  by  the  signs  +  and  —  are,  therefore, 
terms.  -  We  understand,  in  this  definition,  that  an  expression 
within  a  sign  of  grouping  is  considered  as  one  number. 

Thus,  the  expressions  8,  x,  Adb,  3a? -^l/,  axh-^c^  a{x  +  y)^  and 
{x  +  y)-i-{a—b)  are  terms.  The  expression  2a^  —  dab  +  ¥  consists 
of  three  terms.  There  are  two  terms  in  the  expression  3(a  +  &}  + 
5(2c— d).     Name  them. 

An  expression  which  consists  of  one  term  is  called  a  monomial ; 
as  x'^g.     Name  other  monomials. 

An  expression  which  consists  of  tico  or  more  terms  is  called 
a  polynomial ;  as  2a  +  36,  a^  —  2ab  +  5^,  x^—x'^-^x  —  1. 

A  polynomial  of  tioo  terms  is  called  a  binomial;  as  x^—g\' 

A  polynomial  of  three  terms  is  called  trinomial;  as  a*  + 
2a'b'  +  b*. 

Like  or  similar  terms  are  either  terms  which  do  not  differ  at 
all,  or  which  differ  only  in  tlieir  numerical  coefficients.  Other- 
wise they  are  called  unlike  or  dissimilar. 

For  example,  $aa??,  ^ax^  and  7ax^  are  similar.  And  3aa?^, 
5bx^  and  4a^x  are  dissimilar. 

Terms  like  2ax^  y  and  3  bix?  y  are  said  to  be  "  similar  in  x  and  ?/." 

EXERCISE    3. 

1.  State  the  number  of  terms  in  each  of  the  expressions  in 
Exercise  2,  and  name  the  expressions  accordingly. 

2.  Which  of  the  following  terms  are  similar  ?  Which  are 
similar  in  6  ?     In  y'^  ?     In  by-  ? 

2by\     Sx%     7bg\     l\by\     hx^y,     Qab,     ab. 


CHAPTER  II. 

FUNDAMENTAL  LAWS  OP  NUMBERS. 

16.  There  are  placed  here  for  convenience  some  properties 
of  numbers,  known  as  the  fundamental  laws  of  numbers.  These 
laws  are  known  to  be  true  for  the  definite  numbers  in  arith- 
metic. They  hold  equally  well  in  the  extended  number  system 
of  algebra.     They  are  called  : 

(1)  laio  of  order  in  addition^ 

(2)  law  of  grouping  in  addition  ; 

(3)  law  of  order  in  midtiplicatio^i  / 

(4)  law  of  grouping  in  midtiplication  / 

(5)  law  of  distribution. 

We  shall  assume  for  the  present  that  these  laws  are  true. 
They  may  be  easily  shown  to  hold  in  particular  cases.  But 
simply  because  they  are  true  in  particular  cases,  we  may  not 
safely  conclude  that  they  are  always  true.  Rigorous  proofs 
that  these  general  principles  are  correct  regardless  of  the  kind 
of  mimhers  involved  will  be  found  in  the  Appendix. 

17.  Law  of  order  in  addition.  N'umhers  to  he  added  mag  be 
arranged  in  any  order  without  cliangi'iig  the  value  of  the  sum  ; 
that  is, 

a-f6  =  6+a. 

For  example,  2  +  5—5  +  2.     Each  sum  is  7. 
Also,  20|  +  71=7K20|.     Each  sum  is  28. 
This  law  is  extended  to  any  number  of  terms. 
Thus,  7  +  2  +  3-2  +  7  +  3=3  +  7  +  2,  etc. 

16 


FUNDAMENTAL  LAWS  OF  NUMBERS  ,       17 

18.  Law  of  grouping  in  addition.     Numbers  to  be  added  may 

be  grouped^  or  combined^  in  any  manner  without  changing  the 
value  of  the  sum  /  that  is, 

fl  +  6  +  c  =  a  +  (64-c). 

For  example,  5  +  7  +  2  =  5  +  (7  +  2). 
Also,  5  +  7  +  2  =  7  + (5 +  2).     Each  sum  is  14.     See  §  5. 
And  3  +  2  +  9  +  4  =  3  +  (2  +  9  +  4)  =  3  +  2  +  (9  +  4)  =  3  +  (2  +  9)  +  4, 
etc.     Each  expression  equals  18. 

19.  Law  of  order  in  multiplication.  Factors  may  be  ar- 
ranged in  any  order  without  chaiiging  the  value  of  the  product  / 
that  is, 

ab  =  ba. 

For  example,  5x3  =  3x5,  for  each  product  is  15. 
Also,  2x  =  x-2\  ax  =  xa\  etc. 

This  law  may  be  shown  to  hold  for  any  number  of  factors. 

For  example,  4x3x2x7  =  4x2x7x3  =  2x4x3x7,  etc.  Each 
product  is  168. 

Also,  5a?^  =  ^^-5;  3axy  =  aSxy  =  axSy^  etc. 

From  this  law  it  evidently  follows  that  in  any  problem  the 
multiplier  and  multiplicand  may  be  interchanged ;  that  is,  the 
multiplicand  may  be  used  as  multiplier  and  the  multiplier  may 
be  used  as  multiplicand. 

20.  Law  of  grouping  in  multiplication.  Factors  may  be 
grouped  in  any  mangier  icithout  chayigiyig  the  value  of  the  pro- 
duct ;  that  is, 

abc = a{bc)  =  (ab)c  —  {ac)b. 

For  example,  2  3-5  =  2- (3-5).     Each  product  is  30. 

This  law  holds  for  any  number  of  factors. 

2 


18  ALGEBRA 

For  example,  4  x  3  x  9  x  2  =  4  x  (3  x  9  x  2)  =  (4  x  3)- (9  x  2),  etc. 

Also,  3x'a-4y  =  (3-4) (a-x^-^/)  =  i2ax^y. 

Likewise,  2x'Sy'5z''=2-3-5x''y^z^={2-35){x^y^z^)=30xhfz\ 

21.  Law  of  distribution.  The  product  of  one  number  muh 
tiplied  by  the  sum  of  two  or  more  numbers  equals  the  sum.  of 
the  iwodacts  obtained  by  multiplying  the  multiplicand  by  each 
of  the  numbers  in  the  midtiplier  ;  that  is, 

x{a  -f  6  -f  c)  =  jra  +  Jr6  +  jrc. 

For  example,        5  (2  +  3  +  4)  =  5-2  +  5-3  +  5-4. 
Each  exjjression  equals  45. 

By  the  law  of  order  in  multiplication  the  above  principle 
may  be  written 

fljr + 6  jr + c  jr  =  (a  +  6  +  c)  jr. 

For  example,      2a? + 3a^  +  4a?  =  (2  +  3  +  4).^  =  9x'. 
The  student  should  note  that  this  law  enables  one  to  add 
similar  terms. 

Thus  to  add  3x1/^  8a?2/^  2x2/^  and  ^xy\  we  have 

3^?/' +  8a?2/' +  2^?/=' +  5x?/2=  (3  +  8  +  2  +  5)iC2/'=18a^2/^ 

In  general,  to  add  similar  terms  take  the  sum  of  the  numerical 
coefficie7its,  and  to  this  attach  the  commo7i  bases  with  their 
exponents. 

From  the  above  rule,  the  rule  for  subtracting  similar  terms 
easily  follows.  Thus,  since  Zx  +  S^c^  lla;,  then,  by  the  definition 
of  subtraction,  \\x—^x  =  3x.  Since  11  —  8  =  3,  this  suggests 
that  the  subtraction  of  similar  terms  is  accomplished  by  subtract- 
ing the  numerical  coefficients^  and  attaching  to  the  remainder  the 
common  bases  with  their  exponents. 

In  fact,  this  principle  is  included  in  the  Law  of  Distribution 
as  it  is  proved  in  the  Appendix. 

For  example,  to  subtract  4x  from  ITa?,  we  have 
\lx  -  4a;  =  (17— 4)x  =  13a;. 


FUNDAMENTAL  LAWS  OF  NUMBERS  19 

The  use  of  the  Law  of  Distribution  in  arithmetic  may  be 
illustrated  by  the  following  example. 
Multiply  234  by  2. 
We  have  234-2  =  (200  +  30  +  4)-2 

=  200-2  +  30-2  +  4-2 
=  400  +  60  +  8 
=  468. 

EXERCISE  4. 

1.  Show  that  21  +  16  +  9  =  9  +  16  +  21. 

2.  Show  that  5  +  25  +  7-25  +  7  +  5. 

3.  Showthat  3  +  6  +  11  =  3+  (6  +  11). 

4.  Show  that  8  +  3  +  2  +  10  -8  +  (3  +  2)  +  10. 
Find  the  product  of 

5.  5£cy  and  lab.  8.  4x7/  and  J  ahc. 

6.  3a,  2b,  and  5c.  9.  Sa^  and  5^^y. 

7.  7x,  |y,  and  ^z.  10.  2abc,  Sx,  4y,  and  is. 
By  using  the  law  of  distribution,  find  the  sum  of  • 

11.  10a,  2a,  and  3a.  15.  ^ab,  ^ab,  bab,  and  |a^. 

12.  2x,  3ie,  4i»,  and  bx.  16.  2icV',  ^^Y,  and  21a;y. 

13.  4x^2/  and  7yaJ^  17.  ax,  2ax,  Sax,  and  4aic. 

14.  Sxy\  9xy\  and  a;-20y2.  18.  ab\bab\  lab\  and  2a^»^ 

19.  £c,  lOic,  iip,  and  \x. 

Show  how  the  law  of  distribution  is  used  to  obtain  the  pro- 
duct of  .  C!  v.^ 


20.  121x4.      y"         21.  231x3.  22.  12.4x2. 

23.  Divide  2a;  by  2  ;  Sx'y  by  3  ;  ay^  by  a. 

24.  Divide  6a£c^  by  6a ;  by  ^x^ ;  by  ax^. 

25.  Divide  21m^i'^  by  3m ;  by  7?i^ ;  by  mn^. 


20  ALGEBRA 

THE  EQUATION. 

22.  Equations.  As  was  suggested  in  §  3,  an  equation  is  the 
statement  that  two  expressions  are  equal,  or  that  they  have 
the  same  value. 

Examples  of  equations  are  :   a  +  h=b-\-a  ;    2-5  — 1=5  +  4  ;    2x-\- 

The  two  equal  expressions  in  an  equation  are  called  the 
members  of  the  equation.  The  expression  written  at  the  left 
of  the  sign  of  equality  (  =  )  is  called  the  first  member,  and  the 
other  is  called  the  second  member. 

Thus,  in  the  equation  2x-{-y=4:y—x,  2x+y  is  the  first  member^ 
and  4y—x^  the  second  member. 

In  some  equations  the  members  do  not  have  equal  values 
for  all  definite  values  of  the  general  numbers  involved.  i 

Evidently  the  two  members  of  2x  +  y=4y — x  have  not  equal 
values  for  all  definite  values  of  x  and  y.  The  members  are 
equal  when  x  =  3  and  y  =  S  ;  but  are  not  equal  when  x=2  and 
2/=4,  or  when  x  =  1  and  y  =  2. 

Likewise,  the  members  of  2.r  +  5  =  9  are  equal  wheh  a?=2,  but 
for  no  other  value  of  x.  This  equation  is  said  to  "be true,"  or 
"  be  satisfied,"  when  x=2. 

Such  equations  as  these,  equations  which  are  not  true  for  all 
definite  values  of  the  general  numbers  involved,  are  called 
conditional  equations. 

On  the  other  hand,  there  are  equations  in  which  the  mem- 
bers are  equal  for  all  definite  values  which  may  be  assigned  to 
the  general  numbers  involved.  Such  equations  are  called 
identical  equations,  or  identities. 

Thus,  a  +  6  =  6  +  a  is  an  identical  equation  or  an  identity.     It  is 
true  for  any  values  ivhatever  that  a  and  b  may  assume. 
When  a=l,  6=2,  it  becomes  3=3. 
When  a  =  2,  6  =  3,  it  becomes  5  =  5. 


THE  EQUATION  21 

a^-b^  =  {a-vh){a-h),  ^^^^x'+x+i,  and  {a^hf  =  a'  ■\-2ab 
+  6^,  are  all  identities. 

23.  Solutions.  We  saw  in  §  22  that  the  members  of  a  con- 
ditional equation  have  the  same  value  for  only  certain  sets  of 
definite  values  of  its  general  numbers.  Any  such  set  of 
values  of  these  numbers  is  called  a  solution  of  the  equation. 
In  particular,  a  solution  of  an  equation  containing  only  one 
general  number  is  any  definite  value  of  this  mimher  for  which 
the  equation  is  true.  This  definite  value  is  also  called  a  root 
of  the  equation. 

Thus,  8a  — 21=3a  +  4  is  true  when  a  =  5.     Hence  5  is  a  solution. 
To  solve  an  equation  is  to  find  the  definite  value  or  values  of 
the  general    number  for  which  the  equation  is  true. 

24.  Axioms.  In  solving  equations  use  is  made  of  truths 
called  axioms.  An  axiom  is  a  truth  Avhich  is  so  simple  that  it 
may  be  assumed  as  self-evident. 

The  followhig  axioms,  which  are  of  most  frequent  use  in 
algebra,  will  be  needed  in  this  book : 

'     1.  If  equal  numbers  be  added  to  equal  numbers^  the  sums  will 
be  equal. 

That  is,  if  A=B,  then  A4-2=B  +  2. 

2.  If  equal  numbers  be  subtracted  from  equal  numbers.,  the 
remainders  will  be  equal. 

That  is,  if  A=B,  then  A— 2=B-2. 

3.  If  equal  numbers  be  m^idtiplied  by  equal  numbers.,  the  pro- 
ducts  will  be  equal. 

That  is,  if  A-^B,  then  3A=3B. 

4.  If  equal  numbers  be  divided  by  equal  numbers.,  the  quotients 
will  be  equal. ^ 

*  It  is  not  allowable  to  divide  by  0.     See  §  219. 


22  ALGEBRA 

That  is,  if  4A=8B,  then  A=2B. 

5.  Like  powers  of  equal  numbers  are  equal. 
Tiiat  is,  if  A=B,  then  A^^B'^ 

6.  Like  roots  of  equal  numbers  are  equal. 
That  is,  if  A2=36,  then  A=6. 

7.  Numbers  equal  to  the  same  number^  or  to  equal  numbers^  are 
equal. 

That  is,  if  A=:C,  and  B=C,  then  A=B. 

8.  The  whole  of  a  number  equals  the  sum  of  all  of  its  parts. 
That  is,  5=2  +  3=1  +  4=1  +  1  +  1  +  1  +  1. 

25.  At  present  we  shall  attempt  to  solve  only  the  simplest 
forms  of  equations  containing  07ie  general  7iumber. 

The  method  of  solving  any  such  equation  will  be  most  easily 
understood  by  studying  the  following  examples. 

Example  1.     Solve  3x  +  4  =  2x-\-7. 

To  remove  4  from  the  first  member,  subtract  4  from  both  mem- 
bers.    This  gives 

Sx  =  2x-\-3.  Axioms. 

To  remove  2x  from  the  second  member,  subtract  2x  from  botli 
members.     This  gives 

x=  S.  Axiom  2. 

Hence  the  solution  is  3- 

Now,  if  3  be  put  in  place  of  x  in  the  given  equation,  it  becomes 

3x3  +  4  =  2x3  +  7, 

or  13  =  13,  an  identity. 

This  shows  that  the  equation  is  satisfied  when  x  is  3. 

In  solving  this  equation,  we  have,  by  using  axioms,  removed 
those  terms  free  of  x  from  the  first  member,  and  those  terms  con- 
taining X  from  the  second  member.  Observe  that  this  is  the 
general  method  in  the  following  examples. 


THE  EQUATION  23 

Example  2.     Solve  6a— 5= 4a +  1.  -|  $ 

6a— 5  is  5  less  than  ija.     Hence,  to -get  6a,  add  5  to  each  mem- 
ber.    This  gives 

6a=4a  +  6.  Axiom  1. 

To  I'emove  4a  from  the  second  member,  subtract  4a  from  each 
member.    This  gives 

2a =6.  Axiom  2. 

To  get  a,  divide  both  members  by  2.     This  gives 

a  =  3.  Axiom  4. 

Hence  3  is  the  solution. 
Replacing  a  by  3  in  the  given  equation,  we  have 

6x3  —  5  =  4x3  +  1,  an  identity. 

Example  3.     Solve  5(x+l)-\-  Sx=S. 
Destroying  the  group,  we  have 

5x  +  5  +  3x=S.  Law  of  Distribution. 

To  remove  the  term  5  from  the  first  member,  subtract  5  from 
each  member.     This  gives 

5x-[-3x  =  S.  Axiom  2. 

Adding  5x  and  3x,  8x  =  3. 

Dividing  both  members  by  8,  we  have 

x  =  ^.  Axiom  4. 

Hence  |  is  the  solution. 

Replacing  x  by  |  in  the  given  equation,  we  get 

5(1  +  1) +3xf  =  8, 
or  8  =  8,  an  identity. 

To  solve  such  equations  as  the  above  we  may  evidently  use 
the  following  fule : 

(1)  Jiemove  all  signs  of  grouping  hy  use  of  the  law  of  distri- 
hution. 


24  ALGEBRA 

(2)  Hemove  all  terrtis  free  of  the  general  number  from  the  first 
member^  and  all  terms  containing  the  general  7iumber  from  the 
second  member.  A  subtracted  term  is  removed  by  adding  it  to 
both  members.^  and  an  added  term  is  removed  by  subtracting  it 
from  both  members. 

(3)  Unite  the  similar  term^s  in  each  member. 

(4)  Divide  both  mennbers  by  the  coefficient  of  the  general  num- 
ber. 

(5)  Test  the  solution  by  seeing  if  for  the  value  found  for  the 
general  number.^  the  equation  becomes  an  identity. 

EXERCISE  5. 

Solve  the  following  equations : 

^  7a  +  3  =  a  +  21.  9.  4  (r«  I  7)  =  36. 

2.  36  +  4  =  25.  10.  ^  I  ^_1  =  ^. 

2  +  3      6      3 

3.  5c-4-12-3c.  11.  16  (a;  +  2)  +  3  (2jc  +  1)  =  79. 

4.  2a;-5  =  7-£c.  12.  2.:c  +  5  (a;  +  3)  =  3£c  +  27. 
6.  3y-7-=14-4y.  13.  l  +  2£c  =  2  {\-\-1x)-^x. 

6.  5.4  =  3^  +  6.  14.  5(/H3)=A  +  35. 

7.  200P-50-50P  +  250.  15.  4/>-3-/a 

8.  6.5£c  +  3.25  =  15.75-6£c.  16.  5A;-5=X;  +  3. 

26.  The  equation  may  be  easily  used  to  solve  certain  kinds 
of  arithmetical  problems. 

To  solve  such  problems,  the  values  of  certain  unknown  nuni- 
bers  are  to  be  found.  If  some  of  these  unknown  numbers  are 
represented  by  letters  of  the  alphabet,  the  conditions  of  the 
problem  will  lead  to  one  or  more  equations  containing  the  un- 
known numbers.      In  some  problems  there  will  be  but  one 


THE  EQUATION  25 

equation  containing  one  tinknoimi  number.  By  solving  this 
equation  the  value  of  the  unknown  number  is  found. 

It  is  evident  that  the  unknoton  numbers  in  these  problems 
become  the  general  numbers  in  the  equations.  Consequently, 
the  general  numbers  of  any  equation  are  sometimes  called 
unknown  numl)ers. 

The  following  examples  will  show  in  detail  how  the  equation 
may  be  used  to  solve  arithmetical  problems.* 

Example  1.  Of  two  unequal  numbers  the  greater  is  twice  the 
less,  and  the  two  together  equal  99.     Find  the  numbers. 

Let  i»?=the  less. 

Then  2ic=zthe  greater^  for  "  the  greater  is  twice  the  less." 
Hence,  iZJ  +  2a?=99,  for  "  the  two  together  equal  99." 
Therefore  3x=99.  §  21. 

a?=33.  Axiom  4. 

And  %x=m.  Axiom  3. 

Hence  the  less  number  is  33,  and  the  greater  is  ^^. 

Observe,  that  when  x  was  taken  to  represent  one  of  the  num- 
bers, the  next  two  statements  followed  from  the  conditions  stated 
in  the  problem. 

Example  2.  John,  Henry  and  James  have  among  them  30 
marbles.  John  has  twice  as  many  as  Henry,  and  James  has  as 
many  as  John  and  Henry  together.     How  many  has  each  ? 

Let  a=  the  number  of  marbles  Henry  has. 

Then     2a  =  the  number  of  marbles  John  has.  Why  ? 

And      3a  =  the  number  of  marbles  James  has.  Why  ? 

Therefore  a  +  2a  +  3a = 30 .  Why  ? 

Adding  hke  terms,  6a =30.  Law  of  Dis. 

*  Some  authors  prefer  to  represent  unknown  numbers  by  only  the 
last  few  letters  of  the  alpliabet.  Tliere  is  no  good  reason  for  this.  On 
tlie  other  hand  the  student  sliould  be  able  to  consider  the  number 
represented  by  any  letter  in  the  equation  as  the  unknown  number,  and 
to  solve  for  its  value. 


26  ALGEBRA 

Dividing  by  6,  a=5.  ^  Axiom  4. 

Hence  2a =10.        '  Axiom  3. 

And  Sa=15.  Axiom  3. 

Therefore  Henry  has  5,  John  10,  and  James  15. 
The  student  should  see  that  these  numbers  satisfy  the  conditions 
stated  in  the  problem. 


Example  3. 

The  difference  between  the  ages 

of  two  persons  is 

10  years, 

and  the  sum  of  their  ages  is  60  years. 

What  are  their 

ages  ? 

Let 

n=  age  of  younger. 

Why? 

Then 

n  +  10=  age  of  other. 

Why? 

Hence 

n-\-n  +  10=Q0. 

Why? 

Then 

n+n=50. 

Why? 

2n=50. 

Why? 

n=25. 

Why? 

71  +  10=35. 

Why? 

What  are  their  ages  ? 

EXERCISE  6. 

1.  In  a  certain  algebra  class  there  are  24  pupils,  and  there 
are  twice  as  many  girls  as  boys.     How  many  boys  are  there  ? 

2.  I  paid  $7  for  a  pair  of  shoes  and  a  hat.  The  hat  cost 
three-fourths  as  much  as  the  shoes.     AVhat  did  the  hat  costp^ 

Suggestion.  If  you  let  4ic  represent  the  cost  of  the  shoes,  what 
will  represent  the  cost  of  the  hat  ?  If  you  let  x  represent  the 
cost  of  the  shoes,  what  then  must  represent  the  cost  of  the  hat  ? 
Which  is  preferable  ?    Why  ? 

3.  A  certain  man's  age  is  8  times  that  of  his  son,  and  in  ten 
years  it  will  be  twice  as  great  as  the  son's  age  will  then  be. 
What  are  their  present  ages  ? 

Suggestion.  If  you  let  x  represent  the  son's  age  now,  what 
must  represent  the  age  of  the  father  now  ?  What  will  represent 
the  ages  of  each  ten  years  hence  ?    What  equation  follows  ? 


7 


THE  EQUATION  27 

4.  Divide  125  into  two  ^arts  one  of  which  is  35  greater  than 
the  other. 

Suggestion.  If  a?  =  the  smaller  part,  what  must  equal  the 
larger  part  ?  When  x  =  the  larger  part,  what  must  equal  the 
smaller  part  ? 

5.  Find  two  numbers  whose  difference  is  32,  and  one  of  which 
is  3  times  the  other. 

6.  The  sum  of  the  ages  of  two  boys  is  23  years,  and  one  is 
7  years  older  than  the  other.     What  are  their  ages  ?  *  ^ 

7.  A,  B,  and  C  buy  a  piece  of  property  for  $1550.     A  invests  -^ 
twice  as  much  as  B,  and  C  invests  $50  more  than  A  and  B  ^    ^ 
together.     How  much  does  each  person  invest? 

8.  A  farmer  sold  a  number  of  cows  at  $45  each,  and  three 
times  as  many  hogs  as  cows  at  $13  each.  If  all  sold  for  $336, 
how  many  cows  and  how  many  hogs  were  sold  ?  'Z  ♦  1 1— . 

9.  A  rectangular  field  is  twice  as  long  as  it  is  wide,  and  the 
distance  around  it  is  672  yards.     What  are  its  dimensions  ? 

10.  Two  men,  starting  from  points  40  miles  apart,  walk 
toward  each  other,  one  at  the  rate  of  2  miles  an  hour,  and  the 
other  at  the  rate  of  3  miles  an  hour.  In  how  many  hours  will 
they  meet? 

11.  If  a  man  can  row  2  miles  an  hour  in  still  water,  what         / 
must  be  the  speed  of  the  current  of  a  stream,  if  by  its  aid  he     ^ 
can  row  down  the  stream  10  i  miles  in  3  hours  ? 

12.  What  number  increased  by  f  of  itself  will  make  112?  l 

{llint.     Let  4£c  =  the  number.     Why  ?    Also  solve  by  letting 

x=  the  number.) 

/13.  What  number  must  be  added  to  48  in  order  that  twice 
the  sum  shall  be  114  ? 

14.  If  to  a  certain  weight  you  add  its  half,  its  third,  and  8 


28  ALGEBRA 

pounds  more,  the  sum  will  be  twice  the  weight.  What  is  the 
weight  ? 

15.  Divide  63  into  two  parts  whose  difference  is  15. 

16.  The  difference  between  two  numbers  is  7,  and  their  sum 
is  137.     Find  the  numbers. 

17.  The  sum  of  two  numbers  is  80  and  the  greater  one 
exceeds  the  smaller  one  by  16.     What  are  the  numbers? 

J         18.  Find  three   consecutive  whole  numbers  whose  sum  is 
^174. 

'  19.  Two.  men  were  employed  to  dig  a  ditch  630  feet  long. 
One  of  them  dug  an  average  of  45  feet  a  day,  and  the  other  60 
feet  a  day.     How  long  was  required  for  them  to  dig  the  ditch  ? 

^;^20.  A  man  started  on  a  bicycle  to  a  town  25  miles  away. 
He  rode  at  the  rate  of  6  miles  an  hour.  On  the  way  the  bicycle 
broke,  and  he  walked  the  remaining  distance  at  the  rate  of 
^\  2  miles  an  hour.  It  required  6i  hours  to  make  the  whole 
trip.     How  far  was  he  from  home  when  the  bicycle  broke  ? 

—21.  A  started  from  a  place  and  traveled  at  the  rate  of  3 
miles  an  hour,  and  3  hours  later  B  •  was  sent  from  the  same 
place  to  overtake  A.  B  traveled  at  the  rate  of  4  miles  an  hour. 
How  long  was  it  from  the  time  that  A  started  until  B  overtook 
him  ? 

^  22.  A  tank  holding  300  gallons  of  water  has  two  pipes. 
One  lets  in  15  gallons  a  minute,  and  the  otiier  draws  out  12 
gallons  a  minute.  If  both  pipes  are  running,  how  long  will  it 
require  to  fill  the  tank  ? 

23.  A  chain  which  contains  60  links  is  divided  into  three 
segments,  whose  lengths  are  as  3,  4,  and  5.  How  many  links 
in  each  segment? 

Suggestion,     Let  3a;   represent   the    number    in    the   shortest 


THE  EQUATION  29 


segment.     What  then  will  represent  the  number  in  each  of 
the  other  two  ? 


24.  One  number  is  twice  as  large  as  another.  If  I  take  4 
from  the  smaller  and  16  from  the  greater,  the  remainders  are 
equal.     What  are  the  numbers  ? 

26.  Four  men  planned  to  form  a  partnership  and  to  buy  a 
piece  of  property,  but,  one  man  dying,  each  of  the  others  had 
to  invest  $2000  more  than  he  had  planned  to  invest.  What 
was  the  cost  of  the  property. 

26.  Find  the  number  whose  double  exceeds  12  by  as  much 
as  9  exceeds  the  number. 

'  27.  A  man  is  now  seven  times  as  old  as  his  son,  and  in  five 
years  he  will  be  only  four  times  as  old  as  his  son  will  then  be. 
W^hat  are  their  present  ages  ? 

28.  How  far  can  I  drive  into  the  country  at  the  rate  of  4 
miles  an  hour,  in  order  that  by  walking  back  at  the  rate  of  2 
miles  an  hour,  1  may  return  in  9  hours  from  the  time  that  I 
started? 

'  29.  A  boy  bought  equal  amounts  of  two  kinds  of  candy, 
one  kind  at  10  cents  a  pound,  the  other  at  20  cents  a  pound. 
It  all  cost  him  90  cents.     How  much  of  each  kind^did  he  get? 

30.  A  solves  a  certain  number  of  the  problems  in  this  exer- 
cise. B  solves  all  of  those  which  A  cannot  solve  and  twenty- 
two  of  those  which  A  solves.  B  solves  two  more  problems 
than  A.     How  many  does  each  solve  ? 


CHAPTER  III. 

NEGATIVE   NUMBER. 

27.  Let  us  attempt  to  solve  the  equation 

Subtracting  2a;  from  both  members, 

03+7  =  4.  Axiom  2. 

Subtracting  7  from  both  members, 

x  =  4:—7.  Axiom  2. 

What  is  4  —  7?  Evidently  this  equation  has  no  solution, 
unless  we  may  subtract  7  from  4.  This  leads  us  to  the  follow- 
ing considerations. 

28.  Extension  of  fundamental  processes.  Counting  gives  rise 
to  the  series  of  arithmetical  whole  numbers 

1,     2,     3^    4,     5,     6,     7,     8,     9,     10,     11,     12,     etc. 

The  sum  of  any  two  numbers  in  this  series  is  another  one 
of  these  numbers.     Thus  3 +  7==  10. 

The  product  of  any  two  numbers  of  this  series  is  also  another 
one  of  these  numbers.     Thus  2x6  =  12. 

But  when  we  attempt  to  divide,  the  quotient  of  two  numbers 
of  this  series  is  sometimes  another  one  of  these  numbers,  and 
sometimes  it  is  not.  Thus  12-^-3  =  4;  but  10^7  gives  no  one 
of  these  numbers.  However,  problems  which  arise  have  de- 
manded that  we  be  able  to  divide  any  number  in  this  series 
by  any  other  of  the  numbers.     Hence,  since  the  whole  num- 

30 


NEGATIVE  NUMBER  31 

bers  are  insufficient  to  express  all  quotients,  we  indicate  the 
quotients,  and  thus  obtain  a  neio  kind  of  7iimibei\  not  found  in 
the  above  series  of  whole  numbers.  These  new  numbers  in 
arithmetic  have  been  called  fractions.  Thus,  10-r-7=-i^,  a 
fraction. 

The  series  of  numbers  in  arithmetic  has,  therefore,  been 
extended  to  include  fractions  as  icell  as  lohoU  niimhers^  and  this 
has  followed  from  the  necessity   of   making  division  always 


Likewise,  when  we  attempt  to  subtract,  the  difference  be- 
tween two  numbers  of  the  above  series  is  sometimes  another 
number  of  the  series,  and  sometimes  it  is  not.  Thus,  8  — 5  =  3  ; 
but,  4  —  7  gives  no  one  of  these  numbers,  and  we  say  7  cannot 
he  subtracted  from  4-  In  general,  we  say  that  a  number  can 
not  be  subtracted  from  a  smaller  number.  However,  problems 
which  arise  have  demanded  again  that  we  be  able  to  subtract 
any  number  from  any  other  number.  Hence,  v^hen  the  sub- 
trahend is  greater  than  the  yninuend^  we  indicate  the  remainder, 
and  thus  obtain  another  new  kind  of  number.,  not  found  in  the 
old  series  of  arithmetical  numbers.  These  new  numbers  are 
called  minus  numbers,  or  negative  numbers.     Thus,  4—7  gives  a 

negative  number. 

• 

The  series  of  numbers  has,  therefore,  been  extended  in  algebra 
to  include  negative  numbers  ;  and  this  has  followed  from  the 
necessity  of  making  subtraction  always  possible. 

29.  Negative  and  positive  numbers.  One  number  may  be 
subtracted  from  another  by  separating  the  subtrahend  into 
parts  and  subtracting  the  parts  one  at  a  time.  It  follows  that 
we  may  write  4— 7  =  4— 4— 3  =  0— 3.  Hence  it  appears  that  a 
negative  number  may  be  written  to  indicate  the  subtraction  of 
a,  number  from.  zero. 


32  ALGEBRA 

Dropping  the  0  in  0  —  3,  we  have  0  —  3=— 3. 
Likewise,  4-5  =  4-4-1  =  0-1  =  -!; 

4-6  =  4-4-2  =  0-2  = -2; 

4-8  =  4-4-4  =  0-4= -4; 

4-9=4-4-5  =  0-5=-5;  etc. 

It  is  clear  that  a  negative  number,  such  as  —3,  indicates  a 
reserved  subtraction,  there  being  nothing,  when  it  stands  alqne, 
from  wliicli  to  subtract  it.  It  is  in  nature  always  a  subtra- 
hend. 

Negative  numbers  may  be  added,  subtracted,  or  used  in  any- 
other  operation  in  which  other  numbers  may  be  used. 

For  the  sake  of  distinction,  ordinary  or  arithmetical  numbers 
in  algebra  are  sometimes  called  plus  or  positive  numbers.  Also 
for  distinction  in  writing  positive  and  negative  numbers,  the 
positive  or  ordinary  numbers  are  often  preceded  by  the  sign 
+  when  standing  alone.  Thus,  6  is  written  +  6 ;  a  is  written 
+  a.  When  clearness  would  not  be  sacrificed,  however,  the 
sign  +  may  be  omitted  from  the  positive  numbers.  The  sign 
--^^jnust  never  be  omitted. 

When  standing  alone  the  positive  numbers  1,  2,  3,  etc.,  or 
+  1,  +2,  +3,  etc.,  are  read  either  «  plus  1,"  "  plus  2,"  "  plus  3," 
etc.,  or  "  positive  1,"  "  positive  2,"  "  positive  3,"  etc.  And  the 
negative  numbers  —1,  —2,  —3,  etc.,  are  read  either  "  minus 
1,"  "  minus  2,"  "minus  3,"  etc.,  or  "  negative  1,"  "  negative  2," 
"  negative  3,"  etc. 

The  positive  and  negative  numbers  of  algebra  are  called 
algebraic  numbers. 

The  signs  +  and  —  written  before  numbers  are  called  the 
signs  of  the  numbers  or  signs  of  quality.  They  may  always  be 
considered  as  signs  of  operation  i7i  algebra;  and  when  con- 
venient, as  signs  of  quality^  or  distinction. 

Two  numbers  which  differ  only  in  their  signs,  such  as  +  8 


NEGATIVE  NUMBER  33 

and  —8,  are  said  to  have  the   same   arithmetical,  or   absolute 
value. 

30.  Opposite  numbers.  Since  a  negative  number,  such  as 
—  5,  always  implies  a  reserved  subtraction,  when  it  is  combined 
with  an  arithmetical  or  positive  number,  it  tends  to  destroy  of 
the  positive  ntmiber,  a  part  equal  to  itself. 

Thus,  when  9  and  —5  are  combined,  —5  destroys  5  of  the  9, 
leaving  4. 

Accordingly,  positive  and  negative  numbers  are  sometimes 
called  opposite  numbers. 

In  subtraction,  when  the  minuend  and  subtrahend  are  equal, 
the  remainder  is  zero.  This  is  equivalent  to  saying  that  when 
two  opposite  numbers  with  equal  absolute  values  are  combined, 
the  value  of  the  result  is  zero. 

Thus,  3  and  —3  give  0  ;  25  and  —25  give  0  ;  +a  and  —a  giveO. 

31.  Opposite  concrete  magnitudes.  Many  concrete  magni- 
tudes are  capable  of  existing  in  opposite  states^  such  that  one 
tends  to  destroy  an  equal  amount  of  the  other ;  and  hence  their 
numerical  values  may  be  represented  by  positive  and  negative 
numbers.  A  few  are  here  suggested.  The  student  should 
study  these  examples  carefully. 

(1)  Debt  and  credit.  If  I  have  $20  in  a  bank,  and  owe  the 
bank  $20,  paying  the  debt  will  leave  me  nothing.  The  indebt- 
edness destroys  the  credit.  Hence  debt  and  credit  are  opposite 
magnitudes. 

If  I  have  $10  in  my  pocket,  and  make  a  purchase  amounting 
to  $15,1  will  have  left  $10  — $15,  or  — $5.  The  —  $5  means 
that  I  not  only  have  no  money  left,  but  am  $5  in  debt.  In  this 
sort  of  problem,  then,  a  negative  number  means  indebtedness. 
If  we  call  indebtedness  negative^  credit  will  \)Q positive.  Indebt- 
edness will  destroy  credit. 


34  ALGEBRA 

(2)  Forces  in  opposite  directions.  If  two  persons  pnll  in  oppo- 
site directions  on  cords  attached  to  the  sanie  object,  each  with 
a  force  of  100  lbs.,  the  object  will  not  move.  The  two  forces 
have  destroyed  each  other.  If  one  pulls  100  lbs.,  and  the  other 
l^ulls  only  80  lbs.,  the  -effect  upon  the  object  will  be  the  same 
as  if  the  first  person  alone  had  pulled  with  a  force  of  20  lbs. 
If  the  force  exerted  by  one  person  be  represented  by  a^9ost7iy6 
number,  the  other  force  will  be  represented  by  a  negative  num- 
ber. 

(3)  Motions  and  distances  in  opposite  directions.  If  a  person 
walk  100  ft.  east,  then  turn  about  and  walk  100  ft.  west,  he 
will  arrive  at  his  starting  point.  The  result  is  the  same  as  if 
he  had  not  moved  at  all.  The  second  motion  destroyed  the 
effect  of  the  first.  If  the  motion  in  the  first  direction  be  called 
positive,  the  second  motion  will  be  called  negative. 

Accordingly,  distances  moved  through,  or  distances  measured 
in,  opposite  directions  are  called  opposite  distances.  Thus  if 
distances  measured  to  the  right  of  the  point  A  are  called  posi- 

A 

1 . 

-  + 

tive,  distances  measured  to  the  left  of  A  are  called  negative. 
Hence  in  representing  distances,  the  signs  +  and  —  serve  to 
indicate  the  directions  in  which  they  are  measured. 

(4)  Positive  and  negative  temperature.  If  temperature  above 
zero  is  called  positive,  temperature  below  zero  will  be  negative. 
+  10°  means  10°  above  zero.     —10°  means  10°  below  zero. 

From  the  above  illustrations  it  is  seen  that  a  negative  num- 
ber has  the  property  of  oppositeness.  Thus  both  positive  and 
negative  numbers  may  be  represented  by  distances  along  a 
straight  line  measured  from  a  common  starting  point.  If  dis- 
tances to  the  right  represent  positive  numbers,  then  negative 


NEGATIVE  NUMBER  35 

numbers  must  be  represented  by  distances  in   the   opposite 
direction 

—5       ~4        --3        -2       -1  o  1  2  3  4  5    

To  every  point  to  tlie  right  there  corresponds  a  similar  one 
to  the  left.  +  3  is  represented  by  a  point  3  units  to  the  right 
of  some  point,  as  o,  on  a  straight  line.  —3  is  represented 
by  a  point  3  units  in  the  opposite  direction.  If  we  measure  3 
units  in  one  direction,  then  from  this  point,  measure  3  units 
in  the  opposite  direction,  we  return  to  the  starting  point,  ^^e.,  3 
and  -3  =  0.  '  . 

32.  Addition  of  algebraic  numbers.  Addition  was  defined  in 
§  3  as  the  process  of  combining  two  or  more  numbers  into  one. 
We  shall  indicate  the  addition  of  algebraic  numbers  by  writing 
them  in  succession  with  their  signs. 

Thus,  to  indicate  the  addition  of  +6,  +2,  —4,  +3,  and  —5,  we 
write  +6  +  2—4  +  3  —  5.  The  first  term  in  the  expression  being 
positive,  we  may  omit  the  sign,  giving  6  +  2—4  +  3—5. 

Algebraic  numbers  to  be  added  are  sometimes  placed  in  a 
column  with  their  signs.     Thus, 

-3 
+  2 
-6 

Evidently  two  positive  numbers,  since  each  is  an  ordinary 
arithmetical  number,  ina\j  he  combined  by  adding  their  absolute 
values  ;  and  their  sum  will  be  a  positive  number. 

Thus,  5  +  9  =  14. 

Since  the  negative  number  —5  indicates  a  reserved  subtrac- 
tion, then  to  add  —  5  to  any  positive  number  means  to  combine 
it  by  subtraction;  that  is,  to  subtract  5  from  the  number. 
(See  §  30.)  The  same  reasoning  would  apply  to  any  other 
negative  number. 


36  ALGEBRA 

Thus,  the  sum  of  7  and  —5  is  7—5=2.  And  the  sum  of  3  and 
—  5  is  3  — 5=— 2.  Remember  that  when  the  subtrahend  is  greater 
than  the  minuend  the  remainder  is  negative. 

Also,  to  subtract  each  of  two  numbers  in  succession  is 
equivalent  to  subtracting  their  sum.  Hence  two  negative 
numbers,  since  each  is  a  subtracted  number,  inai/  be  combined 
by  adding  their  absolute  values  ;  and  the  sum  will  be  negative. 

Thus,   to  add  —7  and  —5  we  have  —7—5=  — 12. 

From  the  above  considerations  we  evidently  have  the  follow- 
ing rules  of  addition  of  algebraic  numbers : 

(i)    To  add  two  numbers  with  like  signs,  find  the  sum  of  their 
absolute  values.,  arid %>refix  the  common  sign. 
Thus,  7  +  15=22;  -7-15=-22. 

(^)  To  add  t\no  numbers  with  iinlike  signs.^  firul  the  difference 
between  their  absolute  values,  and  attach  the  sign  of  the  arithmet- 
ically greater. 

Thus,  -7  +  15  =  8;  7-15  =  -8. 

{3)  To  add  three  or  more  algebraic  numbers,  2^'^oceed  from  left 
to  right,  performing  each  step  according  to  (1)  or  (2). 

Thus,  to  find  the  sum  of  6,  —9,  3,  and  —7,  we  write  6  —  9  +  3—7; 
6-9  =  -3;  -3  +  3=0;  0-7=-7. 

Another  rule  is  sometimes  to  be  preferred  to  Rule  (3). 
It  is  illustrated  in  the  following  example  : 

7-3-9  +  6-5  +  21  =  -3-9-5  +  7  +  6  +  21  Law  of  order. 

=  (  —  3-9  — 5) +  (7  +  6  +  21)  Law  of  grouping. 

=  _17  +  34.  Rule  (1). 

=  17.  Rule  (2). 

The  same  method  is  applicable  to  any  problem.  Hence  we 
have  the  following  rule  : 

(4)    To  add  three  or  more  algebraic  numbers,  find  the  sum  of 


NEGATIVE  NUMBER  37 

the  positive  numbers  and  the  sum   of  the  negative  numbers  hy 
{1);  then  find  the  sum  of  the  sums  hy  {2). 

Thusin  21-6-3  +  5  +  1; 

21  +  5  +  1=27; 
-6-3=-9; 
27-9  =  18. 

33.  Subtraction  of  algebraic  numbers.  Since  subtraction  is 
the  inverse  of  addition,  the  rule  for  subtraction  of  algebraic 
numbers  may  be  obtained  from  addition. 

In  §  3  we  established  the  principle : 

subtrahend  +  remainder  =  minuend. 

Hence  if  s  stand  for  the  subtrahend  in  any  problem,  and  r 
for  the  remainder,  the  minuend  will  be  r  +  s. 

Now  (r  +  s)  —  s  represents  the  sum  of  the  minuend  and  the 
subtrahend  with  its  sign  changed. 

But  {r-\-s)  —  s  =  r-\'{s  —  s)  by  Law  of  Order  for  Addition. 

By  §  82,  s-s  =  0. 

Hence  r+(s— s)  =  r. 

That  is,  the  sum  of  the  minuend  and  the  subtrahend  with 
its  sign  changed  equals  the  remainder. 

Hence  the  rule :  . 

To  subtract  one  algebraic  number  from  another^  change  the 
sign  of  the  subtrahend^  then  proceed  as  in  addition. 

Thus,  to  subtract     —5  from  10,  we  write 
10+5  =  15,  remainder. 

Again,  6  from  —7  gives  —7—6  =  — 13. 
The  latter  might  have  been  written         —7 

6 

-13 


38  ALGEBRA 


EXERCISE  7. 

Find  the  sum  of 

1, 

7,  -2,  3,  -5. 

9.  From  6  take  —3. 

2. 

6,  5,  -3,  -1. 

10.  From  -6  take  3. 

3. 

3,  -15,  -12,  1,  -9,  10. 

11.  From  -7  take  -5. 

4. 

-6,  -5,  -12,  -8,  4. 

12.  From  3  take  8. 

5. 

12            5     Ql 
2'>  —  ¥»    — IT'   ^2' 

13.  6-(-8)=? 

6. 

A.    -f    -h 

14.  -3-(-7)=? 

7. 

sV'  T6'  — 1»  —  i- 

15.  5-(-3)-(-6)=? 

8. 

5.25,-2.5,  -7.34,  4.35. 

16.   -10-(-4)-(-3)=? 

17.  Since  the  sign  —  always  indicates  a  subtraction,  what 
name  may  be  given  to  any  expression  witliin  a  sign  of  group- 
ing which  is  preceded  by  tlie  sign  —  ? 

18.  From  7-3-6  +  12  take  -6  +  14  +  3-7. 

19.  From  the  sum  of  -3-5  +  2  and  12-6-1  take  the  sum 
of  26  +  111-9  and  3i  +  7-l. 

20.  Is  the  absolute  value  of  an  algebraic  number  always 
diminished  by  subtraction  ?    Illustrate. 

21.  Do  you  ever  add  arithmetically  when  you  are  subtract- 
ing algebraically  ?     Illustrate. 

22.  Is  the  absolute  value  of  an  algebraic  number  ever  de- 
creased by  addition  ?     Illustrate. 

23.  A  balloon  lifts  up  with  a  force  of  200  lbs.  A  weight  of 
150  lbs.  is  attached  to  it.  What  is  the  effect?  If  the  weight 
is  represented  by  a  positive  number,  what  will  represent  the 
lifting  force  of  the  balloon  ?  What  will  represent  the  result 
when  the  weight  is  attached  ? 

24.  A  freight  car  is  running  at  the  rate  of  20  feet  per  sec- 
ond. If  a  man  walks  toward  the  front  on  the  roof  of  the  car 
at  the  rate  of  5  feet  per  second,  how  fast   will  he  be  moving 


NEGATIVE  NUMBER  39 

relative  to  the  ground  ?  If  the  speed  of  the  car  is  called  posi- 
tive, what  algebraic  number  will  represent  the  speed  of  his 
Avalking  ?  What  algebraic  number  will  represent  his  speed  re- 
lative to  the  ground  ?  If  he  walks  toward  the  rear  of  the  roof 
of  the  car  at  the  same  rate,  what  will  represent  the  rate  of  his 
walking?  What  will  be  his  speed  relative  to  the  ground? 
What  will  represent  it  ? 

25.  I  deposit  in  the  bank  at  different  dates  $100,  $175,  and 
$95.  I  check  out  at  different  times  $25,  $10,  $87.25,  and  $37.75. 
If  my  account  is  then  balanced,  how  much  will  I  have  in  the 
bank  ?  If  the  deposits  are  called  positive,  what  will  represent 
the  checks  ?     Show  how  thus  to  find  the  balance. 

26.  On  the  earth's  surface  longitude  west  is  called  positive 
longitude  ;  longitude  east,  negative  longitude  ;  latitude  north, 
positive  latitude  ;  and  latitude  south,  negative  latitude.  How 
could  you  describe  the  position  of  a  town  whose  longitude  is 
112°  east,  and  latitude  75°  north?  In  what  country  Avould  a 
city  be  whose  longitude  was  +90°  and  latitude  +40°  ?  Longi- 
tude -90°  and  latitude  +40°  ? 

34.  Multiplication  of  Algebraic  Numbers.  The  laws  of  signs 
for  multiplication  of  algebraic  numbers  come  directly  from  the 
definition  of  multiplication  given  in  §  3  ;  viz,  to  obtain  the  prod- 
uct of  two  numbers  we  must  use  the  multiplicand  as  we  m,ust  use 
1  (unitf/)  to  obtain  the  multiplier. 

Suppose  the  multiplier  to  be  +3. 

To  obtain  +3  we  must  add  three  I's. 

Thus,  +3  =  1  +  1  +  1. 

Hence  to  obtain  the  product  (  +  5)  (  +  3),  we  must  add 
three  +5's. 

Hence  (  +  5)(  +  3)  =  +  5  +  5  +  5=+15. 

To  obtain  the  product  (  —  5)  (  +  3),  we  must  add  three  —  5's. 


40  ALGEBRA 

Thus  (-5)  (  +  3)  = -5-5-5= -15. 

Suppose  the  multiplier  to  be  —3. 

To  obtain  —3  we  must  subtract  three  I's  from  0  ;  that  is, 
change  their  signs  and  add  them. 

Thus  -3=-l-l-l. 

Hence  to  obtain  the  product  (  +  5)(  — 3),  we   must  subtract 
three  +5's  from  0  ;  that  is,  change  their  signs  and  add  them. 
Thus  (  +  5)(-3)- -5-5-5= -15. 

To  obtain  the  product  (— 5)(  — 3),  we  must  subtract  three 
—  5's  from  0  ;  that  is,  change  their  signs  and  add  them. 

Thus  (-5)(-3)= +5  +  5  +  5= +  15. 

The  preceding  reasoning  will  evidently  apply  to  any  multi- 
plier and  any  multiplicand.     Hence  the  following  laws  : 

(1)  The  product  of  tvio  numbers  loith  like  signs  is  positive. 
Thus  (  +  8)(  +  9)  =  +  72,  and  (-8)(-9)=:  +  72. 

Note  that  all  of  the  signs  +  could  have  been  omitted. 

(2)  The  product  of  tico  numbers  vnth  unlike  signs  is  negative. 
Thus  .(  +  8)(-9)  =  -72,  and  (-8)(  +  9)  =  -72. 

35.  A  product  consisting  of  an  odd  number  of  negative 
factors  is  negative  ;  and  a  product  consisting  of  an  even  number 
of  negative  factors  is  positive. 

This  follows  easily  from  §  34. 
For  (  — 1)(  — 1)=  +  1 ;  Even  number, 

hence  (-1)(-1)(-1)  =  (  +  1)(-1)  =  -1 ;  Odd  number, 

and  (-1)(-1)(-1)(-1)  =  (-1)(-1)  =  +  1;        Evennumber. 
and(-l)(-l)(-l)(-l)(-l)  =  (  +  l)(-l)  =  -l;     Odd  number, 
and  so  on. 
Thus  (-6)(-5)(-2)=-60;  (-3)(-4)(-5)(-6)= +360, 


NEGATIVE  NUMBER  4| 

Since  the  product  of  two  positive  factors  is  positive,  it  is 
easily  shown  by  reasoning  similar  to  the  foregoing  that  a  pro- 
duct consisting  of  any  number  of  positive  factors  is  j^ositive. 

Thus,    (  +  6)(  +  5)(  +  4)  =  +120,    and     (  +  3)(  +  7)(+ 1)(  +  2) 

=  +42. 

36.  Signs  of  Powers.     If  the  factors  of  a  product  are  all  equal, 

the  product  becomes  a  power.      See  §  13. 

Thus,  (-3)(-3)(-3)(-3)=(-3)*;  read  "the  fourth  power 
of  -3." 

Hence  from  §  35  we  get  the  following  laws  : 

(1)  Any  power  of  a  positive  number  is  positive. 

Thus  (  +  2y*  =(  +  2)  (  +  2)(  +  2)=+8;    (  +  4)-^  =(  +  4)  (  +  4)=16  ;  etc. 

{2)  Any  odd  j)ower  of  a  negative  number  is  negative ;  any 
even  power  of  a  iiegative  nu^nber  is  positive. 

Thus,  (-2)^  =(-2)  (-2)  (-2)  =  -8  ;  (-3)^  =(-3)  (-3)  (-3)  (-3) 
(-3)  = -243. 
And(-2)*=(-2)(-2)(-2)(-2)  =  +  16  ;  (-5f  =(-5)  (-5)  =  +  25. 

To  find  the  value  of  a^6^  when  a=3,  6=— 2;  we  have  a^  b^  = 

32(-2)^  =9  (-8)^-72. 

If  a=-3,  6=2,  aW={-3yx2^=9x8=72. 

37.  Division  of  Algebraic  Numbers.  The  laws  of  signs  in 
division  of  algebraic  numbers  are  easily  deduced  from  the  cor- 
responding laws  in  multiplication. 

In  §  3  it  was  shown  that 

divisor  x  quotient  =  dividend. 

Hence,  since- (  +  5)  (  +  3)= +15,  then  (  +  15)^(  +  3)= +5  ; 
since  (-5)  (  +  3)  =  -15,  then  (-15)--(  +  3)  =  -5 ; 


42  ALGEBRA 

since  (  +  5)  (-3)--15,  then  (-15)--(-3)= +5  ; 
since  (-5)  (-3)= +15,  then  (  +  15)---(-3)  =  — 5. 

Manifestly  the  same  laws  apply  here  which  apply  to  multi- 
plication. The  reasoning  in  the  particular  cases  above  will 
evidently  apply  to  any  dividend  and  any  divisor.  Therefore 
we  have  the  following  laws  : 

(1)  If  the  dividend  and  divisor  have  like  signs^  the  quotient 
will  he  positive. 

Thus,(-18)-(-2)=  +  9;^^=+2;^-  +4;  etc. 

{2)  If  the  dividend  and  divisor  have  unlike  signs^  the  quotient 

will  he  negative. 

30 
Thus,  (-21)-(+3)  =  -7;--g=-5;  (-  |)-  |  =  -  |;  etc. 


EXERCISE  8. 

Find  the  product  of 

1.  2  and  -7.  7.  9,  -2,  G,  -5. 

2.  -2  and  7.  8.  5,  -7,  +4,  -2,  -1. 

3.  2  and  7.  9.  -7,  -1,  -1,  -1,  -1. 

4.  -2  and  -7.  10.  -1,  -1,  -1,  -1,  -1,  -1. 
6.  -3,  2  and  -5.  11.  -4,  3,  -2,  0. 

6.  -4,  —3  and  -1. 

Find  the  value  of 

12.  (  +  2f.                 14.  (-4)1  16.  (-2)^  (-3)^ 

13.  (-3)*.                 15.  (-If.  17.  (-ly  (-2)*(-8)». 
If  a=2,  ^>=  — 3,  c^  —4,  find  the  value  of 

18.  a'h\     19.   ahc.     20.   h'c,  21.   a'h'c'.          22.    (-2)(-3)i^. 


NEGATIVE  NUMBER  4.3 


Divide : 

23.  -27  by  3.   ^  ^ 

^    27.  6.25  by  -2.5. 

24.  -27  by  -3.    ^  ; 

28.  8  by  -  |. 

25.  21  by  -7. 

29.  -2|  by  17. 

26.  -  1  by  f . 

30.   +216  by  +12. 

If  a=— 2,  h  =  Z,  c=- 

-4,  find  the  value  of 

31.  "^',-5. 

32.  (v'~-G'yJ)\ 

34.  3c^--2^.         -J^  4 
36.         3  . 

33.  c^--6--«^ 

36.  d'^&-^h\ 

EXERCISES  FOR  REVIEW  (I). 

1.  How  does  algebra  differ  from  arithmetic  ? 

2.  How  is  the  addition  of  three  or  more  numbers  indi- 
cated? 

3.  What  are  the  signs  of  multiplication  used  in  algebra  ? 
Why  may  ah  indicate  the  product  of  a  and  h  but  87  not  the 
product  of  3  and  7  ? 

4.  What  is  the  use  of  the  "  signs  of  grouping "  ?  Illus- 
trate. 

6.  IIow  would  you  evaluate  the  following  expression? 
(l  +  2)(9-4)-8-=-(10-6  +  2)  +  3. 
State  a  rule  for  evaluating  expressions. 

6.  What  is  a  general  number  ?  Illustrate.  How  is  it  rep- 
resented in  algebra  ? 

7.  What  do  we  mean  by  factors  of  an  expression  ?  What 
are  the  factors  of  lahc  ? 

8.  What  is  a  ponder  of  a  number?  What  does  a?  mean? 
What  is  the  3  called?  When  a  =  2  what  is  the  value  of  a*?  of 
a*?  oia'^i 


44  ALGEBRA 

9.  What  is  the  coefficient  of  an  expression  ?  What  is  the 
numerical  coefficieyit  of  lab'^  ?  What  is  the  coefficient  of  b^  in 
the  expression  7a//?  of  a?  of  7b'? 

10.  What  is  a /*7eraZ  expression  ?     Illustrate.  ^o 

11.  When  a  =10,  6=4,  c  =  2,  find  the  value  of  -U  ' 

(a.)  {2a-Zb){4a-3c)~{Sa-Sc-b).       ** 

(6.)  {a  +  br-^^^  +  c(a  +  2b). 

12.  If  a  =  4,  6  =  8,  c  =  3,  r?=2,  and  x  =  b^  find  the  value  of 

(a.)  3  aa5  +  5M-2c'a;  +  2cf?^-M\ 

Aa'  +  ^b'     Sd'{2G'-^db'  +  2x)      d^x 
^^■^        a'b  +      a  {b'-c')  b   ' 

13.  What  are  similar  terms  ?     Illustrate. 

14.  State  t\iQ  ftmdamental  laios  of  nmnbers.  Write  them  in 
symbols.     Illustrate  each  law  with  definite  numbers. 

15.  State  the  laws  used  in  the  following  identities: 
2a^-56^-7c2  =  2-5-7a^-6^c''=(2-5-7)(a26V)  =  70a^6V. 

16.  What  law  is  involved  in 

5a;  +  2a;  +  12£c  =  (5  +  2  +  12)£c  =  19a;? 

17.  State  the  axioms  given  in  this  book.  Of  what  use  are 
they? 

18.  What  is  an  equation?  Distinguish  between  a  condi 
tional  equation  and  an  identity.     Illustrate  each. 

19.  What  is  meant  by  a  root^  or  a  solution^  of  an  equation? 

20.  State  the  steps  in  the  solution  of  the  following,  giving 
your  authority  for  each  step. 

l  +  4a;  =  2(a!  +  l)+3. 

21.  A  tree  60  feet  high  was  broken  at  such  a  point  that  the 
part  broken  off  was  3  times  the  length  of  the  part  left  stand- 
ing ;  required  the  length  of  each  part. 


^ko^-, 


^'^  **'     -^    j^  NEGATIVE  NUMBER  45 

22.  The  greater  of  two  numbers  is  5  times  the  less,  and  their 
sum  is  126  ;  required  the  numbers. 

23.  What  is  a  negative  number  ?  A  positive  number  ?  How 
did  they  originate  ?     Illustrate. 

24.  Why  are  positive  and  negative  numbers  sometimes 
called  opposite  numbers  ? 

25.  If  a  boy  weighing  75  pounds,  is  holding  a  toy  balloon, 
pulling  upward  with  a  force  of  15  pounds,  how  may  these 
numbers  be  represented  by  positive  and  negative  numbers  ? 

26.  In  exercise  25,  if  75  is  called  +75  what  is  the  15? 
What  force  will  be  required  to  lift  the  boy  and  balloon  ? 

27.  If  I  have  $1000  and  owe  11500,  by  what  number  may 
my  financial  condition  be  expressed  ? 

28.  How  else  may  50°  above  zero  and  15°  below  zero  be  ex- 
pressed ? 

29.  What  is  the  difference  between  algebraic  numbers  and 
arithmetical  numbers  f 

30.  How  do  we  add  algebraic  numbers  ?  Find  the  algebraic 
sum  of  -10,  +5,  -7,  -3,  +4,  and  +8. 

31.  How  do  you  subtract  algebraic  numbers?  Illustrate. 
Upon  what  fundamental  principle  is  the  proof  based  ? 

32.  From  the  sum  of  6,  5,  —3,  —1,  take  the  sum  of  ~15, 
-12,  1,  -9,  10. 

33.  State  the  laios  of  signs  in  multiplication.  Upon  what 
important  definition  is  the  proof  of  these  laws  based  ?  By  the 
use  of  this  definition  prove  that  (—2)  (—4)=  +  8. 

34.  Give  the  values  of  (-2)^  (-3)^;  {\f\  (-i)*- 

35.  Give  the  laws  of  signs  in  division.  Upon  what  import- 
ant principle  is  the  proof  of  these  laws  based  ? 

36.  Divide  8  by  -2  ;  -8  by  -2 ;  -81  by  3  ;  -f  by  -|. 


CHAPTER  IV. 

ADDITION  AND  SUBTRACTION  OP  LITERAL 
EXPRESSIONS. 

38.  Addition  of  monomials.  It  was  shown  in  §  21  that 
similar  terms  could  be  added  by  use  of  the  distributive  law 

ax-\'hx-\-cx={a-\-h  +  c)x. 

From  this  law  we  have  the  following  rule : 

To  add  similar  terms^  add  their  coefficients  and  affix  to  this 
sum  the  common  letters  with  their  exponents. 

Thus,  to  add  4x^2/^  —  2a?^2/^  and  —z>x^y^^  we  have,  writing  them 
in  succession  with  their  signs, 

A.x'y^  -  23(^1/'  -  ^x^f = (4  -  2  -  5)  x^y""  =  -  Sx'y^ 

To  add  dissimilar  terms,  write  them  i/t  succession  with  their 
signs. 

Thus,  to  add  2ah,  —^ax^  122/^  and— 32;^  we  have 
2a6-3aa?4-122/'-32;l 

The  addition  of  terms  similar  with  respect  to  certain  letters 
may  be  indicated  by  grouping  the  coefficients  of  the  common 
letters,  and  affixing  to  this  group  the  common  letters  with 
their  exponents. 

Thus,  to  add  2ax^y^  —hx^y,  and  Scoc^y,  we  write 
2ax^y—bx^y  +  Scx^y={2a—b  +  3c)x^y. 
46 


ADDITION  AND  SUBTRACTION  OF  LITERAL  EXPRESSIONS     47 
EXERCISE  9. 


Find  the 

sum  of 

1.       2x 

2.  Za'b 

3.  7c' 

4.  Qabcd 

5.  ax^ 

-3a; 

4a'b 

-be' 

—  21abcd 

^ax' 

5x 

-  a'b 

Qc' 

—  Sabcd 

-  ax' 

—  X 

-lOa'b 

-12c' 

Uahcd 

7  ax' 

6.  bA,  12A,  -3.4,  -7A.  9.  ^pq,  4pq,  -V^pq. 

7.  16P§,  -10P(2,  4P§.  10.  100ylC;-14.4(7,  |^(7. 

8.  4^"^  2^^  -B\  -^B\  11.  ccy,  -9a;V',  4ajV',  -a^y. 

12  p5'r,  —  10^5'r,  ^pqr^  —\pqr. 
13.  ^mhi^^  6^iW,  — 32/?^^^^    — mW. 
Simplify 

14.  ^x—^x—bx^-^x—x.  16.  aWc—2a''¥c^  Q^a^'^c. 

15.  -12y^-3y^  +  4/-72/^         17.  4a^»c-2cia-7^>ac. 

18.  |2/^-3y2;-|2/s  +  |2/2. 
Simplify  by  adding  similar  terms 

19.  4a  +  6a-12a;  +  2a  +  3a5.  ^ 

Solution. 

4a  +  6a-12ic  +  2a  +  3ic=4a  +  6a  +  2a-12.r  +  3£C  Why  ? 

=  (4  +  6  +  2)  a+  (-12  +  3).T 
=  12a-9x. 

20.  xy  —  ab  +  l(iab—\^xij  —  '^ab. 

21.  ^abc''\-2a''bc-babc''^7ah^c-\-7a''bc. 

22.  ^a''  +  W-ba'-b\ 
Indicate  the  sum  of 

23.  Zax^j     bbx\     —7cx^.  24.  2xi/z,     axijz,    -bxyz. 

25.   -7cij\   2y\    Zay\ 

39.  Addition  of  polynomials.  If  all  of  the  terms  of  a  poly- 
nomial are  added  to  an  expression,  the  polynomial  is  added  to 
the  expression.     This  follows  from  the  lavi  of  grouping. 


48  ALGEBRA 

Thus,  if  a,  6,  and  c  are  added  to  a?,  we  have  x-\-a+b-\-c—x+ 
(a+b+c). 

But  x+{a+b+c)  indicates  that  the  polynomial  a+b  +  c  itself  is 
added  to  x. 

Hence,  to  add  two  or  more  polynomials^  write  down  all  of  the 
terms  in  successioii  vnth   their  signs  /   then  combine  the  similar 
terms^  if  any. 
Example  1.     Add  2a  +  3&— 4c,  —4a— 6  + 5c,  and  5a +  6— 2c. 

We  write  2a+36— 4c— 4a— 6+5c+5a+6— 2c=3a+3?)— c. 
The  work  may  often  be  more  conveniently  performed  by 
Avriting  the  similar  terms  in  vertical  columns,  then  adding  the 
terms  in  the  resulting  columns. 
The  above  example  might  be  written 
2a+36-4c 
—4a—  6+5c 
5a+  b—%c 
3a  +  36—  c 

Checks.  In  much  of  the  work  of  algebra  the  student  can  easily 
verify  or  chech  his  results  ;  i.  e. ,  he  may  perform  other  operations 
that  tend  to  show  that  the  first  result  is  correct.  This  is  called 
checking  the  work. 

A  short  method  of  checking  addition  of  polynomials  consists 
of  assigning  particular  values  to  all  of  the  general  numbers  in- 
volved in  the  polynomials  and  in  the  sum,  and  seeing  if  the 
sum  of  the  values  thus  obtained  for  the  polynomials  is  equal  to 
the  value  of  the  sum  of  the  polynomials.  This  is  illustrated 
in  the  following  example. 
Example  2.  Add  6a— 56-f-3c,  7a+106— 6c,  and  8a— 96- 10c. 
Work.  Check. 

6a-  56+   3c  6-  5+   3=       4 

7a  +  106-  6c  7  +  10-  6=     11 

8a-  96-lOc  8-  9-10=-ll 


21a-  46-13C  21-  4-13= 


when 
a=l,  6=1,  c=l. 


ADDITION  AND  SUBTRACTION  OF  LITERAL  EXPRESSIONS     49 

EXERCISE  10. 

Add  and  check : 

1.  x-{'y-\~z^   x—2i/-\-Sz,    —t)x—4ii/  +  z. 

2.  Sa—b  +  2G,    ba  +  c—'2b,    —a  +  Sb—4:C. 

3.  2P  +  4§  +  i?-7.S;    -6P-§  +  3i?  +  2>S'. 

4.  7ac-\-Sxi/,    2ac—lxy. 

5.  20ji?— <7  +  r,    2/)  +  5^— 7r,    — 7p  +  2y  +  3r. 

6.  2a6  — 3*c  +  5ac,    7^c— 2ac  +  6a*,    —  3ac  +  2Z>c. 

7.  a;='  +  i«'  +  a;  +  l,   x'-x'  +  aj-l. 

8.  a^  +  2a^  +  ^>^    a'-2ab  +  h\   2a' -W. 

9.  a;^— y^   a;^  +  6£c^y,    —^xy'^—y^. 
10.  |6«-^^-i-c,    fa-J^    a  +  |$-3c. 


11. 

3£c-'-2£c'  +  a;- 

-4, 

x' 

+  4aj^  +  l, 

3a;^ 

-2a;  +  l, 

x^-x\ 

12. 

-12a;*  +  2a;- 

-1, 

Zx' 

'-2a;^  +  3a^, 

,   a;^ 

+  2a;  +  5, 

Zx'  +  A.x\ 

40.  Subtraction.  The  reasoning  in  §  33  evidently  holds  for 
algebraic  expressions  in  general,  since  any  expression  is  itself 
a  number.  Hence,  to  subtract  one  expression  front  miother^ 
change  the  sign  of  the  subtrahend  and  add  the  result  to  the 
minuend. 

The  sign  of  an  expression  is  changed  by  changing  the  sign  of 
each  of  its  terms.  This  follows  directly  from  Rule  4  in  addition, 
since  an  expression  is  but  the  algebraic  sum  of  its  terms. 

Thus,  changing  signs  in  7—3  we  have  —7+3.  Now  7—3  and 
—7  +  3  are  opposite  in  sign,  but  have  the  same  absolute  value,  4. 

Therefore,  for  subtraction  we  have  the  following  rule : 
Add  to  the  minuend  the  subtrahend  loith  the  sign  before  each  of 
its  terms  changed. 

Example  1.  Subtract  —  4a&  from  —  2a6. 

Changing  the  sign  of  the  subtrahend  and  adding  gives 
-2a6+4a6=2a6. 


50  ALGEBRA 

Example  2.        From  2a— 36  + 5c  take  3a— 2b— 2c. 
Changing  the  sign  before  each  term   in  the  subtrahend  arid 
adding,  we  have 

Work.  Check. 
2a-3h  +  5c                       =4] 
-3a  +  2b  +  2c                      =1  I      when 
a=l,  6=1,  c=l. 

—  a—  6  + 7c  =  5  J 

The  change  of  signs  need  only  be  made  mentally,  the  writ- 
ten signs  of  the  subtrahend  remaining  unaltered.  This  is  illus- 
trated in  the  following  examples. 

Example  3.     Subtract  —2x^  —  4:y^+15a  from  7x^—2y^. 

Work.  Check. 

lx^-2y'                                 =       5  1 
-2a;2- 41/2  +  1 5a                        =       9  I      when 
x*=l,  i/=l,  a=l. 

9xH2i/2-15a  =  -4  J 

Example  4.     From  2a^  +  4^* — 3a7  +  7  take  a^ — 3ar*  +  2x^ — x. 

Work.  Check. 

2x^  +  4cc*  -3a.'+7  =     10] 

x^  -Sx'-{-2x--  X  ^-  1  lwhena;=l. 


af'  +  4x'  +  3x'-2x'-2x  +  7  =     11 


EXERCISE   11. 


1.  From  2  a'b'  take  —SaW.        4.  From  —7abc  take  ^abc. 

2.  From  —  £cy  take  5a»/.  5.  hhy'^—{—%hif)='i 

3.  From  -^aa;Uake  -^ax\       6.  21a;y2-(-3a;y^)=? 

7.  From  Zx—2y-\-lz  take  x^  6y  —  Sz. 

8.  From  7a  +  2a^— 2c  take  8a— 12a^  +  5c. 

9.  From  2x—7y  take  3y— 5a;. 

10.  From  x^—x^+x""—!  take  2x^—x^^x^—l. 


ADDITION  AND  SUBTRACTION  OF  LITERAL  EXPRESSIONS     5I 

n.  From  a^^-Za'b  +  ^ab'-^b^  take  a'-a^h-\-ah'-h\ 

12.  From  lAB^^lxy  —  ZPQ  take  'lxy-V\Pq. 

13.  3ic2-5a;  +  9-(2ec2  +  6£c-4)=?     - 

14.  The  subtrahend  is  x^^-x'-^x'-^-x-^  the  remainder  is  x^^x^ 

+  £c"^  +  l.     Find  the  minuend. 

15.  Subtract  —  4c^s^  +  «*— r^  from  2rt^  +  3i^'^— cV. 

16.  Subtract  \—x^^x^  from  a?^ ;  from  a\  from  0. 

17.  From    the   sum   of    cv'  +  a'b'—d'b'-\'If   and   ^aW—2b'—a' 

take  a^— 5^  +  2£c. 

18.  From   the   sum   of  ^x'—^^x'  +  l  and  ^x' +  p' +  ^x  take  ia;^' 

-x'  +  2. 

19.  From  1.5a-7.2ic'-3.25m='  take  the  sum  of  .4a;^-7.5a  +  5m' 

and  l-.125a  +  3m^ 

20.  From  the  sum  of  a^^— 1,  ce^+aj  +  l,  and  x^—1  take  the  sum 

of  x^  +  a?^  +  2  and  x^  —  1 . 

21.  What  operations  are  indicated  by  3«^  +  2«— (a^— a^  +  «— 1)  ? 

22.  Simplify  ^x^  -\-Sx^+1-  (2x^ -3  +  0^). 

23.  From  7a^  + 3^2-2  take  a'  +  Sa  +  b. 

If    ^  =  2a;^  +  a;-3,   J?=a;='-a;'^  +  2a!  +  2,    C=Sx'-6x  +  l,    find 
the  value  of 

24.  A-B+O.  26.  -A  +  B+O. 
2b.  A  +  B-a                             27:^  A-B- a 

If  a=  -2,  b=~l,  c=3,  f?=2,  find  the  value  of 

28.  2abc  +  ^a'cl  33.  3a -2/;  + 4c. 

29.  b'c  +  adc.  34.  -2ac-2bcl 

30.  a^-5^   .  35.  a'  +  b'  +  c'  +  dK 

31.  2^>c  +  3«(?.  36.  4a'^'  +  2aiV-^>cW. 

32.  a  +  Sd-2c  +  b.  37.  after?- 2a^  +  cr. 


52  ALGEBRA 

41.  Removal  of  signs  of  grouping.  The  negative  sign  (— ) 
always  indicates  that  the  number  following  it  is  a  subtracted 
number.  Hence,  an  expression  inclosed  within  a  sign  of 
grouping  which  is  preceded  by  the  negative  sign  is  a  subtrahend. 

Thus,  in  3a  — (5a— 26),  the  expression  (5a— 26)  is  a  subtrahend. 

But  subtraction  is  performed  by  changing  the  sign  of  each 
term  of  the  subtrahend  and  adding  the  resulting  expression  to 
tlie  min'uend. 

Hence,  a  sign  of  grouping  ^yreceded  by  the  negative  sign  may 
be  rernooed  if  the  sign  before  each  tertn  inclosed  is  changed. 

Thus,  3a-(5a-26)=3a-5a  +  26  ;   -{-Qx-{-5x')  =  6x—5x\ 

The  positive  sign  (  +  ),  preceding  an  expression  inclosed 
within  a  sign  of  grouping,  either  indicates  an  addition  or  serves 
as  a  sign  of  distinction. 

Hence,  a  sign  of  grouping  preceded  by  the  jyositive  sign 
may  be  removed  vnthout  changing  the  sign  of  any  of  the  terms 
inclosed. 

Thus,  4x  +  7.r+(3a?— 2j?+1)=4:c  +  7x  +  3x— 2j?+1. 

Sometimes  signs  of  grouping  are  inclosed  within  other  signs 
of  grouping.  In  such  cases  the  use  of  different  kinds  of  signs 
is  advantageous. 

Thus,       a-{2a-(a— 26)}  ;  x-^y—{2x—y)  +  \;Zx-{^y—x)]. 

In  expressions  of  this  kind,  it  is  best  for  the  beginner  to 
remove  the  innermost  sign  first ,'  then  the  innermost  remaining 
sign  ;  etc. 

Thus,  a-h-{-a-{-h-a-b)) 

=a— 6— .{a— (— 6— a  +  6)},  removing  vinculum, 
=a — 6—  {a  +  6  +  a— 6},  removing  parentheses, 
'  =»a— 6— a— 6— a4-6,  removing  braces, 
=  — a— 6,  adding  like  terms. 
Again,  Ix-Zy-  {(4a— 6)— [5«— 6— (3a:— 2^)— 36]  j 


ADDITION  AND  SUBTRACTION  OF  LITERAL  EXPRESSIONS      5^ 

=7x—:^^J—{4:a—b—[5a—b—3x  +  2y—Sb]} 
=7x—3t/—{4.a—b—5a  +  b  +  Sx—2y  +  Sb} 
=7x—3i/—4a-\-b  +  5a—b—Sx-h2y—3b 
=4x — y  +  a—3b. 

Some  Avork  may  be  saved  by  adding  the  like  terms  as  each 
sign  is  removed. 

EXERCISE  12. 

Simplify  by  removing  signs  of  grouping  and  adding  like 
terms  : 

1.  2a  +  Sb  +  (a-4:b.)  3.  2B'-{B'—4.AC). 

2.  {lx  +  2y)  +  (Zy-2x).  4.  «  +  [2a-(26-r0]. 


5.  x'  +  x'-{x-'lx^)+{x'-{%x''-l'Mx'-x)\, 

6.  a;V-(2.^'^-3.^//)-(2.^^V- {Zxy^-^^f-l^-f^}). 

7.  —(a— {«  —  [«— a— 26]}) 

8.  -(^^  +  2iC"-a5)+(3a;-a;3+l)_(2cc2_^8.T  +  5). 

9.  6a5-- (2y2-4aj^) -7y'  +  (3a;y-2y^) -(3a3'^-4/). 

10.  -[-{-(-,7=^)}]. 

11.  10rt-(3^»-4a)-{2rt-(35  +  «)}-{3/>-(2«  +  6)}. 

12.  2.«-{a?-(-y-a^^^)}.     13.  7- {8-[3  +  (6-2^=a;)]}. . 

14.  —\_x—  {x+  {a—x)  —  {x— a)  —x)  —a~\—x. 

15.  ^(-.(^(^(^(-.1))))). 

16.  10-[16-(14-{12-2}-4)-10]-2. 

17.  Solve  the  equation 

8i«-(5-2^+{3  +  4)=«-(2a;-10). 

42.  Insertion  of  signs  of  grouping.  It  follows  immediately 
from  §  41  that  terms  of  a  pohpioniial  may  he  inclosed  within  a 
sign  of  (frouping^  ichen  this  sign  of  grouping  is  preceded  by  the 
positive    sign,  icithout  changing  the  signs   of  the  terms ;  and. 


54  '  ALGEBRA 

when  preceded  by  the  negative  sign.,  by  changing  the  signs  of  the 
terms. 

Thus,  to  inclose  the  last  three  terms  of  3a?-''  +  2a:^— 4a; +1  in  brack- 
ets preceded  by  the  sign  +,  we  have  3a?^  +  [2x^—40?  + 1].  If  pre- 
ceded by  the  sign  — ,  it  becomes  3^— [— 2a?2  +  4a?— IJ. 

This  principle  is  of  use  In  combining  the  terms  of  a  poly- 
nomial which  are  similar  with  respect  to  some  general  num- 
ber. 

Thus,   combining  the  terms  having  the  same  powers  of  x, 

a£C*  +  &ir^  +  3Ce»+5x*-3iC=*-x  +  4=(a  +  5)x*  +  (6-3)^+(3c-l)x  +  4. 

Again,  4-5^  +  3ca^-ai;c-|-6aa^  +  3.r-7ii;2^4-(a-3)x-|-(3c-7)ar' 
-(5-6a)ar\ 

EXERCISE  13. 

Without  changing  the  values  of  the  expressions,  inclose  the 
last  three  terms  of  the  following  expressions  in  signs  of  group- 
ing preceded  by  the  sign  —  : 

1.  ^—x'-^x^-x'+x-l.  3.  a  +  2^>-3c  +  4(^. 

2.  ax—by  —  cz-Vdw.  f  L(/^»Aj^.  Sd-lOe^bf-g. 

In  the  following  expressions  combine  the  terms  having  the 
same  powers  of  x,  so  as  to  have  the  sign  +  before  each  group  : 

5 .  2x^ — Sx^  +  aoi?  +  b^  +  hx — ex. 

6.  7  +  dx'-^x'—2ax—4ax'  +  Qbx^  +  ^x. 

7.  ax*—l  +  2x^  —  dx*  +  x'^  +  ax^  —  cx  +  dx  —  bx^. 

In  the  following  expressions  combine  the  terms  having  the 
same  powers  of  y,  so  as  to  have  the  sign  —  before  each  group  : 

8.  -y  +  b  +  2y'  +  ay-by\ 

9.  py-qy  +  ry-  sy*  +  2py'  -  Sqy\ 

10.   -Qxif  +  xY-dx'y-2y'-Sy'  +  by. 


ADDITION  AND  SUBTRACTION  OF  LITERAL  EXPRESSIONS      55 

In  the  following  expressions  remove  all  signs  of  grouping, 
and  then  combine  the  terms  having  like  powers  of  x : 

11.  2x-(ax'-bx)+{cx-(2x'-10)\. 

12.  ax'-('2x-6x'-^x). 

13.  (x'-x+l)-{2x-(Sx'-2)-x'}. 

Add  the  following  expressions,  combining  like  powers  of  x : 

14.  a^—1,  ax^  +  bx,  ax^  —  bx^  +  cx,   x-\-b. 

15.  ax^  +  a^x\   2ax—Sx\   ax^  —  %^. 

16.  x^'^—ax^^b,    bx'  +  c,    2x>—d. 

17.  From  a^  +  bx + c  take  bd^  +  gx— d. 

18.  From  2£c^  — 3a;  +  5  take  a—bx^-\^cx. 

1 9.  From  2^^^  ~  2^  take  px^  -{-rx—q. 


CHAPTER  Y. 

MULTIPLICATION  OF  LITERAL  EXPRESSIONS. 

43.  Law  of  exponents.  The  Lxav  of  exponents  in  multiplica- 
tion is  derived  immediately  from  the  meaning  of  an  exponent. 
It  is  understood  here  that  Ave  are  dealing  only  with  exponents 
whidi  are  positive  integers.     The  law  expressed  in  symbols  is 

flr™-a"  =  a"'+". 

WehnYG  a"'  =  aaaa     torn  factors;* 

and  .  a"  —  aaaaa to  n  factors. 

IIe«ice  a"'-a"  —  aaaaa to  7n-j-n  factors. 

m+u 


=  a 
Thus,  d'^a*=aaaaaaa=a' ; 

tA/       tA/     —  eA-/«A/eAytA-/«A/    tAytAy tAj     > 

and  y^-y^=i/+^  =  y^. 

By  similar  reasoning  this  law  may  be  extended  to  any  num- 
ber of  factors.     Hence, 

fl'"-a"-a;'=fl'«+«+/-;  etc. 

That  is,  the  product  of  two  or  more  poicers  of  the  same  base 
is  equal  to  that  base  tcith  an  exponent  equal  to  the  sum  of  the 
exponents  of  the  given  poicers. 

*  Tlie  sign  • is  called  the  sign  of  continuation,  and  means 

"  and  so  on,"  or  "  and  so  forth." 

Thus,  1,  3,  3,  4 is  read  "  1,  3,  3,  4,  and  so  on." 

And  aaaa is  read  "aaaa  and  so  on." 

56 


MULTIPLICATION  OF  LITERAL  EXPRESSIONS  57 

44.  Multiplication  of  monomials.  By  use  of  the  law  of  order, 
the  law  of  grouping,  the  law  of  signs,  and  the  law  of  exponents, 
the  product  of  two  or  more  monomials  may  now  be  found. 

Thus,  to  find  the  product  of  2i)c^y,  —  3ic^2/^  ^"d  7xy^^  we  have 
{2oc^y){—3x^y%7xy^)  =  —2-3-7x^a^xyy^y^,  law  of  order   and  signs, 
=  —  {2-S-7){oc^3cr^x){yy^y^)^  law  of  grouping, 
=—42x®2/*',  law  of  exponents. 

By  these  laws  we  have  the  following  rule  for  the  multi- 
plication of  monomials : 

Find  the  j^roduct  of  the  numerical  coefficients^  using  the  law 
of  signs ;  and  affix,  to  this  the  products  of  the  literal  factors^ 
using  the  law  of  exponents. 

Example.     The  product  of  —  3a&^  7a^x^,  and  — 2&V  is 
( -  3a62)  (Ta^x^)  ( -  2h'x^) = ^2aWx\ 

Note. — The  student  must  be  careful  not  to  combine  the  exponents  of 
different  kinds  of  bases,  as  of  a  and  5.  It  must  be  remembered  also 
that  if  no  exponent  is  written  above  a  base,  the  exponent  1  is  under- 
stood. 

EXERCISE  14. 

Find  the  product  of : 

1.  aW,  and  -a'b'\ 

2.  Qa'b,  -baWand  -2a'b\ 

3.  Sx%  —  2a;y,  —bxg  and  7x\  . 

4.  —iJfy^g'^z^,  —^axg^  and  ^a^. 

5.  \d'h'c'd^  and  -Ua''hc\ 

6.  -'Ia-b\  -4a=^^^c  and  l^c. 

7.  1^,  -lA'B  and  ^A'B\ 

8.  F\  -P'QsLudP'Q'. 

9.  —  ^mW  and  J^mn^. 


68  ALGEBRA 

10.  2.bx\  S.2bxy  and  l.lbxi/. 

11.  12pqr,  —  4/9^$'r^,  jo^^V  and  —r^. 

12.  B''^,  -lOi^/S'^  and  -^BS. 

13.  (a  +  by  and  2(a  +  b)\ 

14.  3«2(^>  +  c)  and  -ba{b  +  cy. 

16.  What  is  the  meaning  of  x-?     Of  »;'»+«?  Of  j^"-*?     Of 

16.  What  is  the  product  of  x"^",  £c"+»'  and  x*"-'"? 

17.  What  is  the  product  of  a^+S  —2a^"  and  --3«? 

18.  What  is  the  meaning  of  {a'f?     Of  (ay?     Of  (a;'')''? 

19.  Find  the  product  of  (aj\  (a'f  and  {a')\ 

45.  Multiplication  of  a  polynomial  by  a  monomial.  The  rule 
for  multiplying  a  polynomial  by  a  monomial  is  obtained 
directly  from  the  law  of  distribution.  Stated  in  symbols  the 
law  is 

(fl  +  6  +  cH- )  x  =  ax-\-bx-\-cx-\- 

If  a  +  ^  +  c+ represents  the  polynomial,  and  x  the 

monomial,  then  ax-^bx-\-cx-^ is  the  product. 

Hence,  the  rule : 

The  product  of  a  polynomial  midtiplied  by  a  monomial  is  the 
si(,m  of  the  products  obtained  by  multiplyi?iy  each  term  of  the 
polynom,ial  by  the  monom^ial. 

Example  1.     Multiply  3ic*— 2a^+6a?— 5  by  4a^. 

(3x*— 2x2  +  6x— 5)4.r^=3x*-4£e— 2x2-4aT^  +  6x-4a^— 5-4a?* 
=  12x'  -  8^  +  24a?*  -  20x^ 

The  work  is  conveniently  arranged  thus  : 
3ic*-2^2  +  6ic-5 
4ic^ 


12x'  -  8x^  +  24x*  -  20itr» 


MULTIPLICATION  OF  LITERAL  EXPRESSIONS  59 

Check.  When  ir=l,  multiplicand=3,  multiplier =4,  product = 
8,  as  it  should. 

Observe  that  since  any  power  of  1  is  1,  substituting  1  for  x  does 
not  check  the  exponents  7,  5,  4  and  3,  but  merely  the  coefficients. 

It  is  always  more  convenient  to  perform  the  multiplication 
from  left  to  right. 

EXERCISE  16. 

Multiply  and  check : 

1.  a'-'lab  +  h'^hy  a'h\ 

2.  cc^— i//  +  a;— 1  by  £c*. 

3.  6«'-5«2^>+25^by  -«^6^ 

4.  -  2a;*  -  ZxY  +  5y *  by  -  Ix^y"-, 

5.  x^y^—m^z^  +  x^w  by  Sxyzw. 
^6.  2pq—Sqr-\~4:rphjbpqr. 

7.  A'-B'  by  AB. 

8.  20x'-dx'  +  2hy—4x\ 

^-  .9.  i  a^y +  1  xyz-^  £cV  by  -xh/z'. 

10.  —ha^h'^&^'^abc—ax—hy—cz  by  —Zabcx^y^. 

11.  I  a;y^— i  «icy  + J  f^ — f  aV  by  —'la^xyK 

12.  Is   there  any   difference  between  4(a— ^)   and    («— ^)4? 
Why? 

Remove  the  signs  of  grouping  and  simplify  : 

13.  x{a—h)  —  (a^h)x. 

Note. — In  these  expressions,  products  preceded  by  the  sign  —  are  sub- 
trahends. Hence,  when  the  signs  of  grouping  in  products  preceded 
by  the  sign  —  are  removed  by  multiplication,  the  signs  of  all  terms 
arising  from  such  products  must  be  changed.  Thus,  — (a +6)  a?  is  a  sub- 
trahend.   Hence  x  (a  —  5)  —  (a  +  h)  x  =^ax  —  hx  —  ax  —  bx 2bx. 

14.  A(2x-Sy)-2(4:X  +  y).  16.  10{x-2y)-(2x-y)S. 

16.  {a-b)c-(a  +  b)c.  —17.   -Sy(xy-x')  +  2x(x'-y'). 


60  ALGEBRA 

18.  -^cJ?h-a¥)-{ah^'-a'b). 

19.  -2(— 2j«*  +  3a;V-y')  +  3(^*  +  2£cV-2y*)- 

20.  3[2a;y-4fc{y-2(a^-2/)}]. 

46.  Product  of  polynomials.  The  product  of  two  polyno- 
mials is  also  obtained  from  the  law  of  distribution.     Thus, 

(ct  +  6  +  c)(£c+y  +  ^)  =  «(^+y  +  2!)  +  i(£c+y  +  2;)  +  c(£c  +  2/  +  s) 

=  ax-\'  ay  -}-  az-{-hx-{-by  -\'hz-\-  cx-\-  cy  -{-  cz. 

Here  the  product  shows  that  each  term  of  the  multiplicand 
has  been  multiplied  by  each  term  of  the  multiplier ;  and  the 
product  is  the  sum  of  all  the  products  thus  formed.  The  same 
method  will  evidently  hold  for  any  two  polynomials.  -  Hence 
the  rule  : 

To  obtain  the  product  of  ttoo  polynomials^  multiply  each  term 
of  the  multiplicand  by  each  term  of  the  multiplier^  and  take  the 
sum  of  the  residting  products. 

Example  1.     Multiply  2x'^—^xy-\-y'^  by  ^x^—xy. 

Write  the  multiplier  below  the  multiphcand  ;  then  multiply 
each  term  of  the  multiplicand  by  3^"^ ;  then  multiply  each  term 
of  the  multiplicand  hj  —xy.  The  similar  terms  obtained  by 
multiplying  should  be  arranged  in  columns  and  added.     Thus 

2aj2-    Sxy  +  y^ 

Sx^—      xy 


(2a^-Sxy  +  y^){Sx^)=    6x'-  dx^'y  +  Sx'y' 
{2x^—3xy  +  y^){—xy)=         —  2x^y  +  3x''y^—xy^ 
6x* — 1  IxTf  +  6x^2/'' — ^if 

Check.  When  x=2  ;  i/=l  ;  multiplicand =3  ;  multiplier =10  ; 
product =30. 

Note. — A  polynomial  is  said  to  be  arranged  according  to  the  powei^s 
of  some  letter  when  the  exponents  of  that  letter  either  increase  or 
decrease  in  the  successive  terms  as  we  pass  from  left  to  right. 


MULTIPLICATION  OF  LITERAL  EXPRESSIONS.  61 

Thus,  x*—2oiy^+x^+x~5,  is  arranged  according  to  the  descending 
powers  of  x  ;  while  ^if+xy^+x^y'^+a^y+x^  is  arranged  according  to 
the  ascending  powers  of  x. 

The  student  will  find  it  an  advantage  in  multiplication  of  poly- 
nomials to  arrange,  if  possible,  both  multiplicand  and  multiplier 
according  to  the  powers  of  some  letter. 

Example  2.     Multiply  x^—2-\-2xhjx  —  ^  +  x^. 
Arranging  both  trinomials  according  to  the  descending  powers 
of  x^  we  have 

x''-\-2x  -2 
x^-^  X  —6 


a?*  +  2ar^- 

2x' 

ie  + 

2x'- 

-  2x 

— 

ex'- 

-12;r+12 

x'*  +  3x^-  6x'—Ux+12 

Check.  When  x=3  ;  multiplicand =13  ;  multiplier =6  ;  pro- 
duct=78,  as  it  should. 

The  product  of  three  or  more  polynomials  may  evidently  be 
obtained  by  multiplying  the  product  of  any  two  by  a  third,  and 
so  on. 

EXERCISE  16. 

Multiply  and  check : 

1.  a  +  b  by  a  +  b.  9.  2a;-5  by  2£c  +  7. 

2.  a  +  b  by  a—b.  10.  4a+7  by  4«-7. 

3.  2x-ij  by  Sx  +  2i/.  11.  -^-10  by  -a^  +  lO. 

4.  2x'  +  by'  by  x—Sij.  12.  Scd+Qx^/  by  2xy—cd. 

5.  x^-\  by  3£c^  +  4.  13.  \x^-\  by  fcc^-i 

6.  4m^-5/^^  by  3?^^  +  2m^  14.  |a  +  |^  by  \a-\b. 

7.  X^pq^^pr  by  2pq—'lpr.  15.  |£c'4-|y'  by  \x—\xj. 

8.  a;  +  3  by  £c  +  2.  16,  2.25a;  +  7.5y  by  -1.5y  +  2.5a;. 


62  ALGEBRA 

17.  .2a-lM  by  4.7a  +  2M.         20.  a  +  b  +  c  hy  x  +  y-\-z. 

18.  ax  +  b}/  +  czhj  ax-hy.  21.  x^—'lx-^^  by  2a;^+5ic-4. 

19.  pq-Vqr—pr  hj pq  +  qr.  22.  £c^  — 1  by  a;^  +  ic  +  l. 

23.  4£c=^-2a;-^  +  3ix;-5by  a;'  +  aj^  +  l. 

24.  £c*  +  a;^  +  l  by  £c*-a;^  +  l. 

25.  a^-}-a^x—ax'^—x^  by  «^— aa;  +  £c^ 
/  26.  2a'^  ba'b'  -  36*  by  a'  -  ^a'b'  +  26*. 

27.  a'-^ab  +  b'hj  a'-ab  +  b\ 

28.  x'  +  x'-i-x'  +  x  +  lhy  x-1. 

29.  J?^-4^C^by  J^^-4i?(7. 

30.  pc+qd—rehy  Sqd—2pc-^re. 

31.  iK^  +  2aa;^-£c*  by  2x'-Sa  +  ax\ 

32.  «-^-3a'6  +  3a6^-6=^  by  a— 6. 

33.  a'x'-2ax'  +  Sa'x-x'  by  aa;-a;^  +  6e^ 

34.  ab  +  bc—cd—bdhjab  —  bc  +  cd+bd. 

35.  a''—aW  +  a'b^  by  a*-6l  39.  ic^'^+^  +  y"-^  by  a;^-2/^ 

36.  aj'^  +  y'^by  aj  +  y.  40.  x''—x''-'  +  x''-''hjx''-i-l. 

37.  X"— y"  by  x^  +  y"".  41.  |t«'— ^  6«6  +  i6''  by  ^a—^b. 

38.  ^=^''  +  6^''  by  a^^'  +  J^^  42.  ^x'-^a'  by  ia^^  +  ^a^ 

43.  ^x'-^x^^  by  2£t— 1. 

44.  i«=^-^a;2  +  ia;— ^  by  3.c^  +  9£c-27. 

45.  f^^  +  |^-iby|a.^-|;^-i. 

46.  1.4i«^-2.7a;  +  3.2  by  2»5''- 1.4a; -3.2. 
47.  x''—y^\)yx^—y'.  48.  £c»+y'^  by  a;™  +  y'". 

Find  the  product  of 

49.  x-1,  iB  +  2,  aj  +  1.  52.  x'  +  x-^l,  x-1,  a;M-l. 

50.  x^-1,  a;2  +  l,  aj*  +  l.  53.  a^-6',  a^-\-b\  a^  +  b\ 

51.  2a-36,  3a  +  26,  a  +  b.  54.  ic  +  1,  aj  +  2,  £c  +  3,  aj+4. 


MULTIPLICATION  OF  LITERAL  EXPRESSIONS  63 

56.  ia^  +  iy,  ix  +  y,  i^  +  iy- 

6  6.  £c» — y %  ic«  +  y ",  x"'  +  y "'. 

57.  f/ -h  ab -}- b'^,  a  —  b,  a^  —  ab  +  b'^,  a-\-b. 

Remove  signs  of  grouping  and  simplify : 
58.  {a-{'b){a  +  b)-(a-b){a-b). 
-    59.  2(Qi'-Sx+l)-(x  +  A)(x-l). 

60.  (x  +  y)(x-2ij)  +  2(x'-y')  —  {x—y)(x+2i/). 

61.  2Xx+l)(x-l)(x  +  2)-4x(l-x)(x  +  S). 

62.  -2a^{a^-3a(a-6)}  +(a'  +  a^>-^>')(«'-6'  +  a^»')- 


CHAPTER  VI. 
DIVISION  OP  LITERAL  EXPRESSIONS. 

47.  Law  of  exponents.  The  law  of  exponents  for  division  is 
obtained  directly  from  the  corresponding  law  for  multiplica- 
tion, by  means  of  the  principle 

quotient  x  divisor  =  dividend. 

Expressed  in  symbols  the  law  is : 

This  follows  from  the  preceding  principle,  for  the  quotient 
«'"-",  multiplied  by  the  divisor  a"  gives  «"*-"«'",  or  a%  the 
dividend. 

Thus,  a^-i-a^=a^-^=a\  This  agrees  with  the  above  principle, 
for  a'^-a^=a^. 

48.  Meaning  of  a".     By  §  47  we  have 

But  «"^a''  =  l,  for  any  quantity  divided  by  itself  gives  1. 

Hence  fl°=l.  Axiom  7. 

That  is,  am/  base  loith  the  exponent  zero  equals  1 . 

Thus,  a^=l  ;  2"=1  ;  10''=1  ;  45"=1;  x^-^x'=3(P=\. 

It  is  therefore  evident  that  if  a  base  appears  to  the  same 
power  in  both  dividend  and  divisor,  it  will  have  the  exponent 
zero,  and  hence  gives  the  value  1,  in  the  quotient. 

49.  Division  of  monomials.  Since  division  is  the  inverse 
of  multiplication,  i.  e^  since  quotient  X  (Umsor  =  dividend^  then 
from  §  44,  the  rule  for  multiplying  monomials,  we  obtain  the 
following  rule : 

64 


DIVISION  OF  LITERAL  EXPRESSIONS 


(;5 


To  divide  one  monomial  by  another^  divide  the  numerical  co- 
efficient of  the  dividend  by  that  of  the  divisor^  using  the  law  of 
signs ;  then  divide  the  literal  ^factors  by  subtracting  the  ex- 
ponents of  the  bases  in  the  divisor  from  the  exponents  of  the  like 
bases  in  the  dividend  to  obtain  the  exponents  of  these  bases  in 
the  quotient. 

Thus,  since  (4a'&V)(-3a*60=:-12a«6V, 

( - 1 2a«65c*)  ^  ( _  3a*6c'0  =       4.a'b*c\ 
which  may  evidently  be  obtained  by  the  above  rule. 
Likewise,         ( -  IGx'y'z')  --  {Sx^y^z')  =  -  2o^z'. 


EXERCISE  17. 


Divide : 

1.  a^  by  a^. 

2.  a'b'  by  a'b. 

3.  -QaW  by  3a' 

4.  lSx^y''z*  by 
6.  4«^6''c^  by  a^bcK 

6.  2Sa*b'c'  by  -7ab\ 

7.  -lOOyyby  -2bp'r/. 

8.  -bOx'a'  by  4x'a\ 

9.  42aWc  by  QaWc. 

10.  \m^ii^p^  by  — \rnn^p^. 


11.  r'sH'  by  —  3rV^. 
13.  -5a;™+i  by  -ic». 


13. 


by  a" 


14.  —  12£C'»+"2/™+"by  Sy'^aj" 

15.  x^'-^'y  by  of-'^y. 

16.  -12s¥by  ^st\ 

17.  18y^"-^  by  2y"-l 

18.  IZz'^-^y"'  by  62"-"^'". 

19.  7i5"'5'"  by  3^"'5". 

20.  lls^+V-^  by  Qs-'-^r'-^' 


50.  Division  of  a  polynomial  by  a  monomial.  If  a  polynomial 
be  divided  by  a  monomial,  the  quotient  multiplied  by  the 
divisor  must  equal  the  dividend.  Hence,  the  quotient  must 
be  such  an  expression  that  the  product  of  its  terms  by  the 
divisor  will  give  the  terms  of  the  dividend.  Therefore,  the 
terms  of  the  quotient  must  be  obtained  by  dividing  the  terms 
of  the  dividend  by  the  divisor.  Hence  the  rule : 
5 


QQ  ALGEBRA 

To  climde  a  polynomial  hy  a  tnonomial^  divide  each  term 
of  the  dividend  hy  the  divisor,  and  take  the  sum  of  the  residting 
quotients. 

Example.     Divide  a?^  4-40?^— 5x*  by  ic^ 

{x^  +  4:Qd' — ^x^)  -^x^=x^  +  4a^ — ^x^. 

The  work  might  be  written 

x^ 4- 4x'-5x'  ^^,    4x'-5x;\ 
x^ 

Another  form  often  used  is 

x^)a^  +  4x^-5x^ 
x*  +  4:X^  —  5x^  ' 


EXERCISE  18. 

Divide : 

1.  x'  +  x'  by  x\  5.  -S0a'b'-27a'b'  by  -SaW. 

2.  xy-xy  +  4:S(^y  by  xy.  6.  Sa'-6a*b  +  9d'b'  by  Sd\ 

3.  x''—bx*^Sx'  by  —x\  7.  a'-d'b-d'c  by  -a' 

4.  ^Ix^-l^x'  by  -9x\  8.  1x'''-\^x'y^  by  ^x\ 

9.  4a;y  +  8a;y-12j«yby2£cy. 

10.  3£c2— |a;y  +  |  a^y  by  fa^. 

11.  — «+5— cby— 1. 

12.  «=^  +  a6  +  ^'by  aW. 

1 3.  ^ficy  +  5£cV'  -  ^a^y  by  —  liaj^'. 

14.  3.25,73' -5.2a;«  +  9.75x^  by  .25i«l 

15.  10a;'»+='-4a;»+3  by  2a!l  17.  y»+V"+^  +  y'"+V'+^  by  a;»+y+». 

16.  a^'^—a"^^  by  a».  18.  x"- +  x''+'' -V  x'^^"  by  a;'. 

5 1 .  Division  of  one  polynomial  by  another.  The  process  of 
dividing  one  polynomial  by  another  is  based  upon  the  principle 
that  the  quotient  multiplied  by  the  divisor  gives  the  dividend. 
The  process  is  best  explained  by  taking  an  example  as  follows, 


DIVISION  OF  LITERAL  EXPRESSIONS  67 

First  arrange  both  dividend  and  divisor  according  to  the  descend- 
ing powers  of  x.  See  §  46.  The  work  may  be  indicated  as 
below. 

Dividend                          x*+  a^  +  7x^—&x  +  8a^  +  2x-\-8    Divisor. 
{x'^-\-2x  +  8)x^=  x^  +  2x^  +  8x^ of—  x+1    Quotient. 


—x'—  af—Qx  +  S 
{x^-\-2x  +  8)-{—x)=  —af'—2od'—8x 

x^  +  2x  +  8 
(x^4-2x  +  8)-l=  '  a;^  +  2a?+8 

Now  the  product  of  the  term  of  highest  power  in  x  in  the 
quotient  and  the  highest  term  in  the  divisor  must  give  the  highest 
term  in  the  dividend.  Hence,  the  highest  term  in  the  quotient  is 
the  quotient  obtained  by  dividing  the  highest  term  of  the  dividend, 
X*,  by  the  highest  term  of  the  divisor,  x^.  This  gives  x^^  the  first 
term  of  the  quotient. 

Multiply  the  whole  divisor  by  the  term  of  the  quotient  just 
found.  This  gives  x*  +  2x^  +  8x'^,  which  is  placed  below  the 
dividend. 

The  dividend  is  the  product  of  the  divisor  by  the  whole  quo- 
tient. And  X*  +  2x^  4-  8x^  is  the  product  of  the  divisor  by  the  term 
of  the  quotient  found.  Hence,  subtracting  this  from  the  dividend, 
the  remainder  —x^—x^—Qx-\-8  must  bo  the  product  of  the  divisor 
and  the  part  of  the  quotient  to  be  found. 

Therefore  the  product  of  the  next  highest  term  of  the  quotient 
by  the  highest  term  of  the  divisor  must  equal  the  highest  term  of 
the  remainder.  Hence,  dividing  —x^  of  the  remainder  by  x'^  of 
the  divisor  gives  — x,  the  second  term  of  the  quotient. 

Multiply  the  whole  divisor  by  the  new  term,  —x  ;  subtract  the 
product  from  the  remainder.     This  leaves  oc^  +  2x+8. 

Evidently  the  third  term  of  the  quotient  will  be  obtained  from 
this  second  remainder  just  as  the  second  term  was  obtained  from 
the  first  remainder. 

By  continuing  this  process,  all  of  the  terms  of  the  quotient  may 
be  found. 


68  ALGEBRA 

The  above  reasoning  will  evidently  apply  to  any  dividend, 
divisor,  and  their  quotient. 

If  the  divisor  is  an  exact  divisor  of  the  dividend,  the  work 
may  be  carried  on  until  a  remainder  zero  is  found.  Otherwise, 
the  process  may  be  continued  until  a  remainder  is  obtained  in 
which  the  highest  term  is  of  lower  power  than  the  highest  term 
of  the  divisor.     This  is  a  true  remainder. 

It  is  evident  that,  in  the  latter  case,  the  dividend  is  com- 
posed of  the  remainder  and  the  product  of  divisor  and  quo- 
tient.    That  is, 

dividend  =  quotient  x  divisor  +  remainder. 

If,  in  the  preceding  example,  we  had  arranged  both  dividend  and 
divisor  according  to  the  ascending  powers  of  x^  we  would  have 
obtained  the  same  result,  except  that  the  order  of  the  terms 
would  have  been  reversed. 

We  have,  therefore,  the  following  rule  for  dividing  one  poly- 
nomial by  another : 

Arrange  both  the  dividend  and  divisor  according  to  the  de- 
scending or  ascending  powers  of  some  letter. 

Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor  to.  obtain  the  first  term  of  the  quotient. 

Multiply  the  lohole  divisor  by  this  term  of  the  quotient^  and 
subtract  the  result  from  the  dividend. 

Treat  the  remainder  as  a  neio  dividend,  {being  careful  to 
arrange  the  terms  as  before)  and  repeat  the  process,  continuing 
until  either  the  ronainder  zero,  or  a  true  remainder,  is  found. 

Example  1.  Divide  a^— 11a +  30  by  a— 5. 


a^- 11a +  30 
a^—  5a 

—  6a +  30 

—  6a +  30 


a  — 5 

a—^  Quotient. 


DIVISION  OF  LITERAL  EXPRESSIONS  09 

Check.  When  a=l  ;  dividend =20;  divisor  =—4;  quotient  =—5; 
as  it  should. 

Example   2.    Divide  789ify'  +  45xy^  +  Uy*  +  Ux*  +  45a^y  by  2x'  + 
7y^  +  5xy. 


Uoc*  +  45a?*2/  +  78ic22/2  +  4:5xy^  +  Uy* 
14x*  +  35^  +  49.rV 


2xM-  5xy  +  72/^ 


7^^;=^  +  5xy  +  2i/'' 


lO.r'2/  +  29xY^  +  45it?2/"^  +  14y* 
10x'y  +  25x'y'  +  S5xy' 

4:X^y'^  +  10xy^-{-14y^ 

4;rV  +  10a^y'  +  14</* 

Here  the  terms  are  arranged  according  to  the  powers  of  x 
without  reference  to  the  formation  in  y. 

Check.  When  a?=l,  i/=l;  dividend=196;  divisor=14;  quotient 
=14,  as  it  should. 

Note.  Where  the  error  may  be  in  the  exponents,  other  values  than  1 
sliould  be  used  for  the  letters  or  general  numbers.  See  "  observation" 
§45. 

Example  3 .  Divide  2x'' — 7a?  +  a?*  + 1 0 — 7x'^  by  x''  +  2— Sx. 

a?*  +  2a?^— 7a?2— 7£C  +  10  |a?2— 3x  +  2 


oc*—33(^  +  2x'^  \x^-^5x  +  Q 


IX' 

\x' 


5x'—  9x^—  7.r  +  10 
5x^-15x''  +  10x 

6ir*-17a^4-10 

6ar^-18;y+12 

X—  2^  True  Eemainder 

Check.  When  x=S;   dividend=61;    divisor=2;    quotient=30; 
rem.  =1. 

Observe,  that  either  a?=l  or  a?=2  would  reduce  the  divisor  to 
zero,  hence,  these  values  could  not  be  used.     See  §  219. 

Example  4.  Divide  x^  +  y'^—z^-\-3xyz  by  x  +  y—z. 
Here  each  remainder  should  be  arranged  with  the  term  or 
terms  containing  the  highest  power  of  x  preceding  all  others. 


70 


ALGEBRA 

ar*                    +Sxyz  +  y^—z^ 

x  +  y-z 

\  sc^  +  x'^y-x^z 

x'^—xy  +  xz  +  y'^  +  yz  +  z^ 

—x^y  +  x^z              Sxyz                                       T   ^ 
^    -x'y          -xy'+  xyz 
x'^z  +  xy^  +  2xyz 
>    x'^z           +  xyz—xz^ 

xy^+  xyz  +  xz^-Vy^ 
^  xy""                    -vy'-y^z 

xyz  +  xz"^        +y''z—z^ 
.  xyz                 -vy^z-yz^ 

xz^                +yz'^—2^ 
[^  3dz^                +yz'^—z^ 

EXERCISE  19. 

Divide  and  check : 

1.  Grt^-7a-3  by  2a-3.  3.  17.y  +  2y^  +  21  by  3  +  2y. 

2.  lQx  +  bx'  +  ^  by  £c  +  3.  4.  5a-^  +  lla  +  2  by  a  +  2. 

5.  3a^-7a-2-2a^  by  1  +  a. 

6.  a;*-2a;y  +  y*by  ic'-2a;y  +  2/l 

7.  c^-lOc  +  24  by  c-6.  11.  aj^'  +  y^^  by  x  +  y. 

8.  a'  —  (^'  by  a  +  b.  1 2.  ic'  +  y ^  by  aj  +  y. 

9.  a'-h'  by  a-^>.  13.  ««-^»«  by  ci'-h\ 
10.  a;*-.y*  by  x-y.                      14.  a«  +  ^»«  by  a='  +  5^ 

15.  144x^-1  by  12a;  +  l. 

16.  a;^  +  3a;2  +  3a;+lbya;^  +  2ic  +  l. 

17.  03*— 2ic^y  +  2a!y— y*  by  a;^— 2/1 

18.  a;^  +  a!*  +  l  by  aj^-a^^  +  l. 

19.  aj^-1  by  a?-l. 

20.  7aV-3a*-5aV  +  3aaj«-2a;«  by  a;?  +  2aaj^— aj*. 

21.  2a*-9a^  +  17a^-14abya'-2a. 

22.  x^-y^hy  ^-y\ 


DIVISION  OF  LITERAL  EXPRESSIONS  71 

23.  iz;*-12£c=^  +  54£c^-108a;+81  by  x^-Qx+9. 

24.  x'+x'  +  l  by  x*  +  x'  +  l. 

25.  15/  +  13y-17/-3  by  6/  +  3-4y. 

26.  x'-a^hj  x'  +  2x'a  +  2xa'  +  a;\ 

27.  l-x-Sx'-x'  by  l  +  2a;  +  £fl 

28.  x'  +  2a;=^y2  +  9y*  by  x'  -  2xi/  +  3yl 

29.  a3*  +  81  +  9£c^  by  Sx-x'-9. 
ZO.  a'-Sb'-l-Qabhya-l-2b. 

31.  a^"^— 41a^-120  by  x'-h^x  +  b. 

32.  ^—\f^2yz—z^  by  x  V y—z. 

33.  x^—x^y—xy^-\-y^  by  x-  Ty^  —  2xy. 

34.  a-' -243  by  a- 3. 

35.  13«^^  +  71t«-70a^-20  +  6(«'by  3a'^  +  4-7a. 

36.  i6''-'-y'  by  cc^'— y. 

37.  a*  +  2«^-8a  +  12-7a^  by  a^  +  2-3«. 

38.  £c^  +  y^  +  2=*  — 3£cys  by  £c  +  y  +  ^. 

39.  ic=^  +  3xV  +  ^xy^-{'  y'  +  2'  by  a^  +  y  +  s. 

40.  \a''\-^^ab'  +  ^^h'  by  ia  +  i^. 

41.  ^i^a^»-32^^  by  \c(}-2b. 

'   ^%  ^-^x'-\-\xY  +  y'hy  ^x^  +  ^xy^f. 

43.  \xY-\-Th^\^l\x^  +  \xy. 
^4.  6«"'"*  +  3a^"*6*"  +  a'"^"''*— ^=^'"  by  (r  +  ^'". 

45.  £c*"— y*"  by  x'+y". 

46.  12,x''*+^  +  8£c'^— 45iK"-^  +  25£c»-^  by  6a;- 5. 
•  47.  -^\x^^  +  ly^''  by  ^aj^u  _|_  i.yu^ 

48.  4a2  +  4a^  +  <^^-12ac-6^>c  +  9c2  by  2a  +  ^-3c. 

52.    The    fraction.       An    indicated    quotient    is    called    a 
fraction.     The  terms  used  in  arithmetic  are  also  applied  to 


72  ALGEBRA 

algebraic  fractions.  In  a  fraction,  the  dividend  is  called  tlie 
numerator,  and  the  divisor,  the  denominator.  The  numerator 
and  denominator  are  called  the  terms  of  the  fraction. 

A  fraction  may  be  expressed  by  any  of  the  signs  used  to  ex- 
press division. 

a      ,                        x^  +  1 
Thus,  ,—  a/6,  a-v-6,  a :  6,   -'  are  fractions. 

'  h,     '  '  x+1 

Any  laws  that  apply  to  quotients  must  evidently  apply  to 
fractions. 

In  the  following  sections  a  few  principles  are  established 
concerning  fractions  that  will  be  needed  in  the  subsequent 
work.     For  the  full  treatment  of  fractions  see  Chapter  XI. 

53.  Since  quotient  X  divisor  =  dividend,  it  follows  from 
the  preceding  definitions  that 

fraction  X  denominator  =  numerator ; 

a 

that  is,  J--  b  =  a. 

This  is  a  useful  principle. 

54.  The  product  of  tioo  or  more  fractions  equals  the  product 
of  the  7iumerators  divided  by  the  product  of  the  deno)ninators  ; 

a    b    c       abc 

that  is,  ,  — = . 

X  y   z      xyz 

To  establish  this,  call  the  product  p ;  i.  e.,  let 

abc 
-.-.-=p. 

xyz 

Multiply  both  members  of  this  assumed  equation  by  x,  y 
and  z  in  turn. 


DIVISION  OF  LITERAL  EXPRESSIONS 

Multiplying  both  members  by  x  we  have 


Then 


or 


Then 


a 

X-  • 
X 

b 

y 

c 

z 

=px. 

'     a 

b 
y 

c 

z  ~ 

=px 

h 

y 

c 
z 

•  a  = 

=px. 

bers  of  this 

equa 

h 

c 

z 

a  = 

=pxi/. 

z 

•  a  = 

=2)xi/ 

or 


a  b=pxy. 


^3 

Axiom  3. 
§  63. 

Law  of  order, 
y,  we  have 

Why? 

Why? 

Why? 


Then  multiplying  both  members  by  z^  we  have 

cab=pxyz^  or  abc=pxyz. 
Now  dividing  both  members  by  xyz^  gives 

abc 

xyz 

abc     abc 
X   y  z      xyz 
since  each  member  equals ^x 

This  reasoning  can  be  extended  to  any  number  of  fractions 


P- 


Therefore, 


Axiom  4. 
Axiom  7. 


Example  1.  — —  k-^  •  -^  =  ^  ,.  ., — ■. 

5     2a^    a*       5-2a^  a 


Example  2. 


2^2 


3x^ 


3z/ 


42/'     -2/' 


32/-42/^-(-2/3) 


60^ 
12/- 


55.    To  dlolde  any  number  by  a  is  equwaUnt  to  muUiplying 
the  number  by  the  fraction  — ;  that  is, 


rV 


^4:  ALGEBliA 

n         1 
~  =  n. — 

a         a 

For,  since    quotient    X   divisor  =   dividend^   to   multiply    - 

by  a  gives  ~a  =  n.      And  to  multiply  n-  by  a  gives  n—a  = 
n-  {  --a]  =nl^n,  for  the  the  same  reason.     Therefore  -  and 

?i-must  each   be  a   quotient  obtained   by  dividing  n  by  «, 

and  hence  must  be  equal. 

Thus,  184=J/=3. 

56.    The  law  of  distribution  holds  also  for  di vision  ;  that  is, 
a-\-b  +  c  _a     be 

X  XXX 


For,"  +  ^'  +  ''  =  («  + j  +  c)i,  by  §  55, 


=  a-  +  /'>>-  +  c-,by  law  of  distribution, 

XXX 

a      h      c 
=  -  +  -  +  -,  by  §55. 

XXX 


CHAPTER  VII. 
POWERS  AND  ROOTS. 

57.  Involution,  The  definition  of  a  power  of  a  number  was 
given  in  §  18,  and  the  laios  of  signs  ofpoioers  were  established 
in  §  36.     Tlie  student  sliould  now  reread  those  two  sections. 

The  process  of  raising  a  base  to  any  power  is  called  involution. 
The  following  laws  of  exponents  will  now  be  established  for 
involution,  wliere  the  exponents  are  assumed,  of  course,  to  be 
positive  integers. 

58.  Power  of  a  power. 

(a")"'  =  fl"™. 

That  is,  the  mth  povter  of  the  nth  poioer  of  any  number  equals 
the  nmth  povner  of  that  number. 
For,  by  definition  of  an  exponent, 

(«»)'"  =  a"  •  a" -te" to  m  factors, 

:^^«+n+n+ m  terms,  ^^^   ^f  CXpOUeUtS,    §    43. 

Thus,  {a^)^=a^-a^-a^-a^=a''^  ;  (x^)^=aj^-'=a;2^ 

59.  Power  of  a  product. 

{aby  =  a'b'. 

That  is,  the  nth  poioer  of  the  product  of  tioo  numbers  equals 
the  product  of  the  nth  powers  of  those  numbers. 

For,  by  the  meaning  of  an  exponent,  * 

{abY  =  ababab to  n  factors, 

75 


76  ALGEBRA 

=  {aaa to  /i  factors)  (^^^ to  ?*  factors). 

Laws  of  order  and  grouping. 

By  similar  reasoning  the  law  can  be  shown  to  hold  for  any 
number  of  factors. 

Thus,  {xyzwy=xYz'iv';  {2aby=2^aW=8aW; 
{-3aWy={-3yia'y{b'y=81aW\  by  §  58. 

Combining  the  laws  of  §58  and  §59,  we  get  the  following 
rule  : 

To  raise  a  monomial  to  a  required  poioer^  raise  the  numerical 
coefficient  to  the  required poicer^  using  the  laws  of  si(/?is;  then 
inultiply  the  exponent  of  each  literal  factor  h\j  the  exponent  in- 
dicating the  required  power  y  then  indicate  the  product  of  the 


Example.     Raise  —  Sa^V^^^  to  the  third  power. 
We  have  (-5a.-V^2)'=(-5)'^*V'^'''=-125^^2^V. 


60.  Power  of  a  fraction. 


\b]  "F 


That  is,  the  nth  power  of  a  fraction  equals  the  nth  poicer  of 
the  numerator  dioided  by  the  nth  poioer  of  the  denominator. 

For,  It)  =  ----- to  n  factors,  by  def.  of  a  power, 

\oJ        h  b  h 

a  aa •  •  to  7i  factors  e  c^ 

§  54, 


Thus, 


-hbh 

to  n  factors' 

a^ 

-3xVV 
2d'b'  J 

(-3^V)'       81^1/1^ 

POWERS  AND  ROOTS  77 

EXERCISE  20. 


Raise  to  the  indicated  powers  : 

1.  (ay.  13.  (-ba'xyy 

2.  (ay.  14.  (Smhy. 

3.  (-aWy.  15.  (-2d'x'y. 

4.  (-xyy.  .a^ 


21 


22.  u-^^: 

5.  (xYzy.  *"•  V^V  23.  (ay. 

6.  (axy)«.  ^^    /^'Y  24.  (a^hy\ 

».  (     ^^2/).  18.   ^— ^^).  26.  (-x^Y'^^K 

9.  (7a^yni 

10.  (2.y)l  19.   (-1^).  27.   (^). 

12.  (-xyy.  2«-  ^7-wV-  28.  (^)  . 

6 1 .  Square  of  a  binomial. 

(a  +  6)^  =  a^  +  2a6  +  6^ 

That  is,  the  square  of  a  blnotnial.  equals  the  square  of  the  first 
term^  plus  two  times  the  product  of  the  tico  terms^  plus  the  square 
of  the  second  term. 

For,  (a-^hy  means  (a+^)(«  +  ^).  By  actual  multiplication 
this  becomes  a'^  +  2ai  +  ^^ 

Note. — It  is  understood  here  that  the  symbols  a  and  h  represent  any 
terms  whatever.     Either  term  may  be  positive  or  negative. 

Thus,  Zx^—2y  is  of  the  form  a  +  &,  where  a  stands  for  3a?^  and 
6  stands  for  —2y. 

Example  1.  Square  2x-{-3y. 

{2x+Syy={2xy  +  2{2x){3y)  +  {Syy=4x'-{-12xy  +  Qy\ 


78  ALGEBRA 

Check.  When  x=2  and  y=l;  base  =7;  po\Yer  =49,  as  it  should. 
Example  2.  Square  5x*— 2?/^ 

{5o(f-2y'y={5x'y  +  2(5sc'){-2y')-h{-2yY=25af-20x'y'  +  4y'. 

Check.  When  x=2  and  y=S  ;  base  =22  ;  power  =484,  as  it 
should. 

Examples.  Square —2aH 66. 

(-2aH66)2=(-2a=^)2  +  2(-2a'')(66)  +  (66)' 
=4a'-24a^b  +  3Qb\ 

Check.  When  a=l,  h=2;  base  =10;  power  =100,  as  it  should. 

Example  4.    {(a  +  6)  +  l}'=(a  +  6y'  +  2(a  +  6)  +  l 

=a' +  2a6  +  ^'-^  +  2a  +  26  4- 1. 

It  is  observed  that  since  the  square  of  any  number  is  positive, 
the  terms  obtained  by  squaring  the  terms  of  the  given  binomial 
are  always  positive.  The  other  term  is  positive^  if  the  terms  of 
the  given  binomial  have  like  sigtis,  and  negative.,  if  they  have 
unlike  signs, 

EXERCISE  21. 

Write  out  the  following  squares  by  the  above  rule : 

1.  {x  +  y)\  11.  {^x'-^y.  21.  {m'~^)\ 

2.  (2a  +  J)^  12.  {x-Viy.  22.  {mn-2y)\ 

3.  i'Za^Uy.  13.  (2i«  +  3)l  23.  (46^-5)1 

4.  {2x^-\-yy.  14.  (3ic=^  +  4)l  24.  {x'-l)\ 

5.  {a-bf.  15.  (5«^'  +  l)l  26.  {d'-l)\ 

6.  {x-y)\  16.  (a*  +  10)l  26.  {x'-yy. 

7.  {^x-Zy)\  17.  {la^-^-lxf.  27.  {x'  +  xy. 

8.  {a'-by.  18.  (Sa^'  +  Sa^^)^  28.  (^y-7)^ 

9.  (2a;^-2/'/.  19.  (a;y  +  2a)l  29.  (£c'-10)^ 
10.  (aj*-5/)l  20.  (m-3)l  30.  {x'-by. 


POWERS  AND  ROOTS  79 


50.     U  + 


36.  (2x-Shjy. 

37.  (2ab-{-4:bcy. 


31.  (a;*-a;^)l  41.  (x^'-iy. 

32.  (2a;*-' +  0^)^  42.  (x"  +  iy.  """   V*  '  ^/ 

33.  {6x'-xy.  43.  (cc"-l)l  51^  (-,-~X 

34.  (7a  +  2^.)l  44.  (^._2^»)2.  '    \*'     2/7- 
36.  (3aV  +  ^T-  46.  (2.»  +  3a»)l  ^2.   (^l  +  l^. 

46.  (a^+'-n'^-^y.  53. {(2a +  ^) +  6-} I 

38.  (-2^xyy.  ^7-  i^^^^'  +  ^-^^y-  54.   {3a  +  (5-c)}- 

39.  (-a^i/^  +  a^V)^  ^^'  («"*"-!)'•  56.   {4-(2a  +  ^.)}^ 

40.  (.t'"+1)1  49.  (a"5»-^-a"-'^«)'-  56.   {(4-3/>)-3c}^- 

62.  Square  of  a  polynomial. 

(a  +  6  +  c)^=fl^  +  6'  +  c'+2fl6  +  2flc  +  26c. 
By  actual  multiplication,  it  will  be  found  that 

(a  +  5  +  c)'  =  a^  +  ^'  +  c\+  2ab  +  2ac  +  2bc ;  also 
{a  +  b-\-c  +  dy  =  a'  +  b'  +  c'  +  d'  +  2ab  +  2ac  +  2acl 
-^2bc  +  2bd+2Gd; 
and  so  on,  for  any  number  of  terms. 

That  is,  the  square  of  any  polynomial  equals  the  sum  of  the 
squares  of  all  of  its  terms,  plus  two  times  the  product  of  each 
term  into  all  of  the  terms  following  it. 

Example  1 .  Square  2x^  +  3a?  +  5.  '" 

(2a?^  +  3.^+  5)2= (2ic'0=^  +  (3a^)2  +  (5)2  +  ^{^x'^Zx)  +  2{2x^){S>)  +  2(3if)(5). 
= 4a?^  +  9a;2  ^  25  +  12x^  +  %W  +  30^. 
=4x^12x^  +  29x^  +  30.^  +  25. 

Check.  When  x=l;*  base=10;  power=100.  To  check  ex- 
ponents also  let  X  equal  some  other  number  than  1.  Check  when 
x=2. 


go  ALGEBRA 

Example  2.  Square  a*—2a^  +  3a^—^a. 

{a*-2a'  +  3a'-4ay={ay  +  {-2a'r  +  {Sa'y  +  (-4.ay  +  2{a')(-2a^) 
+  2{a'){Sa')+2(a*){-4a)  +  2{-2d'){3a')-{- 
2(-2a^)(-4a)  +  2(3a'-^)(-4a) 
=aH4a«4-9a*4-16a2-4aH6a«-8a5-12a^  +  16a*-24a3 
=a«-4a^  +  10a«-20a5  +  25a*-24a^  +  16a^  ' 

Check.  When  a=l;  base=— 2;  power=4,  as  it  should.  Let 
a=2  and  check. 

Note. — The  student  should  learn  to  write  out  these  values  without 
indicating  the  work  as  in  the  first  step.  He  should  always  check  his 
work. 

Example  3.    Write  out  (S—2x  +  x''y. 

{S-2x  +  x''y=9-\-4x^  +  x*-12x  +  6x''-4x^ 
=9-12x+10x^-4x^-{-x\ 

Check.     Whena?=2;  base=3;  power=9. 


EXERCISE  22. 

Write  out  the  squares  of  the  following  polynomials  : 

1.  l+x  +  x\  10.  4x'-a'  +  Sax. 

2.  a*  +  a'  +  2.  11.  Sx'-^y"-]  bb-a\ 

3.  ^x  +  Sx'  +  4x\  12.  b'-4ac+10. 

4.  x'+x'  +  x  +  l.  13.  ab-hc\cd-ad. 

5.  2iK*-3a3^  +  4.  14.  a'-^¥-&-d\ 

6.  a-2J  +  3c-4f7  H5e.  15.  2.^•^ - 5a;  +  7 - 3al 

7.  l_a;-|-a;2_|_a;3  4-a;4.  16.  \-x-Vx^—x^-^x'-x\ 

8.  5a;•'-2  +  7«^  17.  ic"  +  y«  +  ^". 
'^,"x-\-y—z-\-w.  18.  a?" — y"+^4-2«+^ 

19.  Show  that  the  rule  of  §  61  is  a  special  case  of  §  62. 

20.  Show  also  how  §  62  could  have  been  obtained  from  §  61 
by  so  grouping  as  to  form  a  binomial. 


POWERS  AND  ROOTS  81 

63.  Any  power  of  a  binomial.  By  actual  multiplication  it  is 
found  that 

(a-{-by  =  a'+3a'b  +  Sab'  +  b'; 
(a  +  by=a'  +  ^a'b  +  Qa'b'  +  ^ab'-}-b'; 
(a-^b)'=a'-^5a'b  +  10a'b'  +  10a'b'-{-5ab'-hb'; 
(a-^by=a'-}-Qalb  +  15a'b'  +  20a'b'-\-16a'b'-\-Qab'-\-b'; 

and  so  on. 

Now  by  comparing  these  few  values  of  the  different  powers 
of  a+b,  it  is  found  that  they  all  may  be  written  out  by  the 
following  laws : 

(i)  The  first  term  in  each  case  is  a  vnth  an  exponent  equal  to 
the  exponent  of  the  binomial ;  the  last  term  is  b  with  the  same 
exponent. 

(^)  The  expo7ient  of  a  in  each  term  after  the  first  is  less  by  1 
than  its  exponent  in  the  preceding  term,  b  appears  to  the  first 
poy>er  in  the  second  ter^n.,  and  its  exponent  in  any  term,  after  the 
second  is  greater  by  1  than  its  exponent  in  the  preceding  term. 
The  sum  of  the  exponents  of  a  and  b  in  any  term  is  the  same  for 
all  terms.,  and  equals  the  exponent  of  the  binomial. 

{S)  The  coefficient  in  the  second  term  equals  the  exponent  of 
the  first  term,  'jind  the  coefficient  of  any  term  is  obtained  from 
the  preceding  term  by  multiplying  the  coefficient  of  term  by  the 
exponent  of  a  and  dividing  the  product  by  a  number  greater  by  1 
than  the  exponent  of  b  in  the  term. 

{4)  The  number  of  terms  is  always  greater  by  1  than  the  ex- 
ponent  of  the  binomial. 

Note.— These    laws    constitute    what    is   known    as  the  Binomial 
Theorem.    This  theorem  was  first  estabhshed  about  the  year  1665  by 
the  great  mathematician,  Sir  Isaac  Newton. 
6 


82  ALGEBRA 

In  chapter  XXITI  this  theorem  will  be  proved  to  hold  for 
any  positwe  integral  exponent,  and  its  application  to  fractional 
and  negative  exponents  will  be  shown. 

Example  1.  Write  out  the  value  of  {x-\-yf. 
By  (1)  and  (2),  the  terms  without  the  coefficients  will  be 
ixf"     x'y     ixfy'^     scr'if     x^y^     x^y^      x^y^     xy^      y^ 
By  (3),  the  coefficients  will  be 

1         8         28         56         70         56         28         8         1 
Hence,  {x  +  yf=j(^  +  ^x^y  +  2%x^y''  +  mx>y^  +  H)x'y^  +  56xV'^  +  28ic'7/« 

+  ^xy''  +  if. 

Check.     When  x=\,  y=l  ;  base=2  ;  power=:256. 

Example  2.     Write  out  the  vahie  of  {x—yf. 

The  exponents  and  coefficients  may  be  calculated  at  once. 

{x-yf=x'  +  ^x\-y)  +  ^x{-yy  +  {-yf 
= x^ — 3x^y  +  3xy^ — y^ . 

Check.     When  x=3,  y=l  ;  base=2  ;  power=8. 
Example  3.     Write  out  the  value  of  (2x^—3y-^y. 
{2x'-3fy=(2xy  +  4{2xy{-:iy')  +  ()(2xy(-3yy  +  4(2x')(-:iy'y  + 
(-3yy      =16x^-96x^y^  +  216xY-21Qx'y^  +  SU/\ 

Check.     When  x=2,  y=l ;  base=5  ;  power=625. 

EXERCISE  23. 

Write.out  the  values  of  the  following  powers : 

1.  (x  +  yy.  8.  (Ax'-Syy.  15.  (f  x— fy^. 

2.  (x~yy,  9.  (x  +  iy.  16.  (ix'  +  Wy- 


3.  (x-ay.  10.  (1  +  ay.  ^^    /a     c 

4.  (x  +  yY\  11.   (x-iy.  '    \^     ^^ 


5.  (^x  +  2yy.  12.  {x-2y.  18.   (^-2 

6.  (^x^  +  yr-  13.  (1+yy.  /I^  ,  M' 

7.  (x'-yy.  14.  (x  +  ^yy,  ^'  \a'^  bj' 


POWERS  AND  ROOTS  83 

20.  (2x-yy.  22.  (2a-^y.  24.  (f«-f^»)«. 

21.  {x  +  i^y.  23.  (i-2a)«.     "  25,{ix-yy 

ROOTS. 

64.  If  all  of  the  factors  in  a  product  are  equal,  one  of  the 
factors  is  called  a  root  of  the  product. 
Thus,  a  is  a  root  of  aaaa,  or  a\ 
The  nth  root  of  an  expression  is  one  of  its  7i  equal  factors. 

Thus,  the  square  root  of  an  expression  is  one  of  its  two  equal 
factors. 

The  cube  root  of  an  expression  is  one  of  its  three  equal  factors. 

The  fourth  root  of  an  expression  is  one  of  its  four  equal  factors. 

The  fifth  root  of  an  expression  is  one  of  its  five  equal  factors  ; 
and  so  on. 

To  indicate  a  root  of  an  expression  the  radical  sign  (|/  )  is 
used,  ^/x  represents  the  fourth  root  of  x.  Here  4,  the  num- 
ber placed  above  the  radical  sign,  is  called  the  index  of  the 
root.  The  index  of  a  root  of  an  expression  indicates  what  root 
it  is,  or  the  number  of  equal  factors  in  the  expression. 

The  index  of  a  square  root  is  usually  not  written. 

Thus,  f/x  represents  the  cube  root  of  x;  i.e.,  one  of  the  three 
equal  factors  of  x.  ya  represents  the  fifth  root  of  a.  ya  is 
the  same  as  j/d.  y  81  represents  one  of  the  four  equal  factors  of 
81  ;  i.e.,  y  81=3. 

A  root  is  called  an  even  root  if  its  index  is  an  even  number, 
and  an  odd  root  if  its  index  is  an  odd  number. 

Thus,  yd  represents  an  even  root  ;   yd  an  odd  root. 

It  follows  from  the  above  definition  that  to  find  the  nth  root 
cf  (i  gimn  number  is  to  find  a  second  number  whose  nth  power 
equals  the  given  number ;  that  is, 

{y^ay  =  a. 


84:  ALGEBRA. 

Thus,  since  5^^=25,  therefore  |/25=5  ;  since  {a}f=a^,  therefore 

The  process  of  obtaining  a  root  of  an  expression  is  called 
evolution. 

6  5 .  Laws  of  Signs.     The  laws  of  signs  of  roots  are  obtained 
from  the  laws  of  signs  of  powers. 
The  following  principles  are  true  : 

(J?)  A  2^ositwe  number  has  at  least  two  even  roots  which  differ 
only  in  signs.  For,  if  two  numbers  have  the  same  absolute  value, 
but  differ  in  sign,  their  like  even  powers  are  equal  and  positive. 

Thus,  since  (  +  3)^=9  and  (-3)^=9,  therefore  v'^9=+3  or  -3. 
Also,  since  (  +  2a^)*=16a^2  and  (— 2a=')*=:16a'^  therefore  y'T^'^ 
+  2a^  or  -2a^ 

The  two  even  roots  of  a  positive  number  are  often  written 
together,  by  use  of  the  double  sign  ± .  Thus  ]/25a*=  ±  5a^,  means 
]/25a*=  +  5a^  or  —  5a^ 

{2)  Any  jjositive  nnmher  has  at  least  one  odd  root.,  which  is 
also  positive  /  and  any  negative  nnmher  has  at  least  one  odd  root, 
which  is  also  negative.  For  any  number  has  the  same  sign  as 
any  odd  power  of  itself. 

Thus,  since  (  +  3)=^  =  +  27  and  (-3)=^= -27,  therefore  ^27=  +  3 
and  ^^^=-3. 


Since   (  +  2)^= +  32   and  (-2)'^=-32,    therefore   |/32=+2  and 


5/ 


V  -32=-2. 

(3)  A  negative  number  has  no  even  root  that  can  be  expressed 
as  a  positive  or  negative  number.  For  any  even  power  of  a 
positive  immber,  or  of  a  negative  number,  is  positive ;  i.  6.,  no 
positive  or  negative  number,  raised  to  an  even  power,  can  give 
a  negative  number. 


POWERS  AND  ROOTS  "  85 


Thus,  y/  —9  is  neither  +3  nor  -3,  for  (  +  3)'= +  9  and  (-3)" 
=  +  9. 

The  indicated  even  root  of  a  negative  number  is  called  an 
imaginary  number,  and  does  not  belong  to  the  series  of  num- 
bers with  which  we  are  now  acquainted.  Imaginary  numbers 
will  be  discussed  in  Chapter  XIV. 

66.  Root  of  a  power.  A  root  of  a  j^otrer  of  a  base  equals  that 
poioer  of  the  base  vnhose  exponent  ^.9  tJie  quotieiit  obtaiued  by 
dioidlng  the  given  exponent  by  the  index  of  the  root.     That  is, 

n  — 77 

ya"'  =  a"'^". 
For,  by  §  58, 

(a'"-5-") "  =  a"'-5-''x»  =  a"'. 

m 

Hence,  i/o^^^"'^'*  or  <^"- 

Thus,  j'/^o  :^  ct2o  -  4  ^  ^5  .  ^^  =  x^  =  x'  ;  f{x-yf=(x-yy^-= 
{x-yf. 

67.  Root  of  a  product.  The  nth  root  of  a  product  equals  the 
product  of  the  nth  roots  of  its  factors  ;  that  is, 

y  ab  —  yay  b. 

For,  by  §  59, 

^yay'br  =  {{raY-{yby=-ah. 
Hence,  ^}  ^  ==  ^^  ~ .  ^«  ^. 

Thus,  ^^^  =  -,^^l>^;  -i>^Y^=  |/^|/^=a^Y;  y"  -  32a?^  = 
y^^y'¥^=-2-x'  or  -2x\ 

68.  Root  of  a  fraction.  The  nth  root  of  a  fraction  equals  the 
nth  root  of  the  numerator  divided  by  the  nth  root  of  the  denomi- 
nator ;  that  is, 


V\ 


S6 

ALGEBRA 

For,  by  §  60, 

Hence, 

^    *     yb 

Thus,  |/g  = 

^''    y    2436^^          ^'2436^'^ 

2a« 
36=^' 

69.  If  the  value  of  the  mdicated  root  of  a  rational  expres- 
sion can  not  be  exactly  obtained,  the  indicated  root  is  called  a 
surd.  An  expression  which  contains  one  or  more  surds  is  called 
a  surd  expression,  or  an  irrational  expression. 

Thus,  |/3,  ]/  c?,  -y/a  +  b,  are  sia^ds ;  Vx—\/y'^  24-i/5,  ixvQ  irra- 
tional expressions,  yl  +  y^  is  not  a  surd  since  1  +  ^/3  is  not  a 
rational  expression. 

A  perfect  ntla.  power  is  an  expression  whose  ni\\  root  can  be 
obtained ;  i.  e.,  the  ?^th  root  of  which  is  a  rational  expression. 

Thus,  since  (it^— 4)2=x*-8a^'  +  lG,  then  yiJc^—%x'-\-U=x'-4.. 
Hence,  a?*— 8a?^  +  16  is  Si  perfect  square. 

70.  Roots  of  monomials.  Roots  of  monomials  may  be  extracted 
by  means  of  §  65,  §  66,  §  67,  §  68. 

Example  1.  Find  the  square  root  of  25a^y*. 

We  have  |/25aV=  V25\/a^i/y*  §  67. 

=  ±.5aV.  §65,  §6G. 

Example  2.  Find  the  fifth  root  of  ~B2x'Y^. 
We  have  ^-32x'Y'=  \/~^^  y'x^^Vy^  %  67. 

=  -2icy.  §  65,  §  66. 

si     125a;V 
Example  3.    Find  the  value  of    \~  21 6a'>&'>' 


./   I2r»y_     rm£i-  ^^  j„ 


4 


216a»&8  f  216  a«6« 


POWERS  AND  ROOTS 


87 

§67. 


-"""^2*  §65,  §66. 


J^Y5- 


Example  4.     Find  the  value  of  i/f+ 
Adding  tr.c«on..  l/ff^.y/S 


=  / 


16^2 
25 


,      =±i|/2,  surds. 
EXERCISE  24. 


Find  the  value  of 

Al  ^9Z^5^                  11.    tM6^5«.  J^^    i/J^, 

2.  |/T6-^y.          ^12.  ^>8k,Tvy^.  '      /^l^l 

4.  v/225m'»«'.       /^U.  ^"=«Tpy?^.  23.  r5?^«. 


^24.  j;/i?5^ 


^ 


^5.  if 27^^«.  15.  i:^32^ 

6.  f-^xY\  '16.  ^/-243«^^y.    *^^"'**   ^^    "^ 

7.  r^^^.  n.  i^-,^V^a^6-A26.  V^T^. 

8.  f/-125m^Vi«.  ,18.  i^e^^^s.        \    26.  i/TT^T. 

9.  iTGlSy:  19.  ^-Vl^aH'^    \  27.  ^/fT]. 
10.  i^ip^l  /20.  v'afy^            \28.  Vf^. 

71.  Square  roots  of  trinomials  by  inspection.  In  §61  it  was 
shown  that  the  square  of  a  binomial  was  a  triuoraial.  I'lie 
square  root  of  a  trinomial  that  is  the  square  of  a  binomial  can 
be  found  by  inspection. 


88  ALGEBRA 

Note. — It  is  not  always  possible  to  extract  any  root  of  any  expres- 
sion.    See  §  69. 

Since  the  square  root  of  the  trinomial  is  to  be  a  binomial,  it 
must  take  the  form  a  +  h.  Plence,  the  trinomial  must  be  of  the 
form  {a-^b)\ov  ce  \'lah-\^h\  From  the  form  of  a'  +  2ah-^b\ 
we  have  the  following  : 

A  trinomial  is  a  perfect  square  if  tioo  of  its  terms  are  perfect 
squares  (a^  and  6^),  and  the  other  term  is  tvnce  the  product  of 
t?wir  square  roots  (2a 6). 

Thus,  a*— 6a2  +  9  is  a  perfect  square  ;  for  |/c?  is  either  a^  or  —a^, 
|/ 9  is  either  3  or  —3,  and  twice  the  product  of  two  of  these 
values,  a^  and  —3,  or  —a^  and  3,  gives  the  other  term  —  6a^  In 
fact  a*— 6a''' +  9  is  the  square  of  a^— 3,  or  of  —  a^  +  3.  Why  not 
select  a'^^- 3  or  -a='-3  ? 

From  the  type  form, 

fl^^-2a6^-6^  =  (a^-6)^ 

we  have  the  following  rule  for  obtaining  the  square  root  of 
the  trinomial : 

Write  the  sum  of  the  square  roots  of  the  terms  that  are  p>er- 
feet  squares^  using  such  signs  that  twice  the  product  of  the  result- 
ing terms  will  gim  the  other  term  of  the  trinomial. 

Example  1.     Find  the  square  root  of  a;^— 14a? +  49. 

We  have  j/p=±ic,|/49=±7'.  ^  Since  the  product  —  14.:r^  is 
negative,  the  terms  must  have  unlike  signs.  Hence  we  use  ^x, 
and  —7  or  —x  and  +7.  Therefore  y  x'  —  \^.x\ ^^—x—1  or— ^  +  7. 
See  §65,  (1). 

Check.     When  x=\  ;  base=36  ;  root=  — 6,  or  6. 

Example  2.     Find  the  value  of  |/9x*  +  30x'-^?/  +  25  z/^. 

Here  |/ 9x*=  ±  3a?',  |/25p=  ±  5?/  Since  +  ^^x^y  is  positive,  we 
must  use  like  signs.  Hence  |/9^*T30^pT252/^=3a?H5?/  or 
-3ic2-52/. 

Check,    When  a?=l,  2/=l  ;  base^^Oi  ;  root=8  or  —8. 


POWERS  AND  ROOTS  g^ 

EXERCISE  25. 

Determine  which  of  the  following  expressions  are  perfect 
squares : 

1.  4a'  +  4ab  +  b\  8.  a'b'-2a'bc'  +  c\ 

2.  a'-^ab  +  9b\  9.  {x'  +  x'  +  l. 

3.  lQx'  +  24x}/  +  9if.  10.  l--ia  +  ^i-a2 

4.  x  +  2xij-\ri/,  11.  25xY-  +  20axij  +  4a\ 

5.  9r«^  +  24a&-16^>l  12.  121aj«-20ic^  +  l. 

6.  m^-10m?2  +  25?zl  13.  «*  +  50a"'  +  625. 

7.  //  +  16^yV  +  64c\  14.  a^-4a/y^  +  4^>l 

15.  IQx'-Ux  +  SQ. 
Find  by  inspection  the  square  roots  of  the  following  trino- 
mials : 

16.  ii;'  +  10a;  +  25.  30.  25a''  +  10a'x  +  x\ 

17.  x'  +  12x  +  SQ.  31.  a'x'  +  2axy  +  7/\ 

18.  a;^  +  16a;  +  64.  32.  9.^y-24«^Z.%  +  16rt*6«. 

19.  CC2  +  18.T  +  81.  33.  l-6a;  +  9a;l 

20.  a3--20i^  +  100.  34.  1^.2  +  ^^^^  +  ^^2^ 

21.  ic^-30ic  +  225. 

22.  a^  +  50«  +  625. 

23.  4a;^  +  28a!-f49. 

24.  9^^ -30a; +  25. 

25.  l6a2-48«  +  36. 

26.  81a;^  +  36£c^  +  4. 

27.  121aj«-22a;M-l. 

28.  lQ9a'  +  U2ab  +  4db\ 

29.  9a;*-12ajy  +  4y. 

—  40.   -^ 

144 


35. 

9^ 

-v^v 

'+¥2/*- 

36. 

a' 

%•- 

2/^' 

37. 

2-  +  1. 
2/ 

38. 

16a;^ 
49 

-2  + 

49 

39. 

4  +*  +  «* 

2  +  ^^^. 

^  1 

iC« 

90  ALGEBRA 

72.  Roots  of  polynomials  by  inspection.  Since  the  process  of 
finding  the  ?^th  root  of  an  expression  consists  of  finding  a  second 
expression  whose  nth  power  is  the  first  expression,  the  roots  of 
some  polynomials  may  be  found  by  the  aid  of  §  63. 

Note. — A  general  process  of  finding  the  square  roots  and  cube  roots 
of  polynomials,  and  of  arithmetical  numbers,  will  be  found  in  the 
Appendix.     These  methods  by  inspection  will  suffice  for  the  present. 

Example  1.     Find  the  cube  root  of  x"*  —  3x'^y  +  3xy^  —  y^. 

Here  the  first  and  last  terms  are  perfect  cubes,  and  there  are 
four  terms.  This  suggests  that  the  given  expression  may  be  the 
cube  of  a  binomial.  See  §  63,  (1)  and  (4).  Taking  the  cube  roots  of 
the  two  terms  which  are  perfect  cubes,  we  get  x  and  —y.  Their 
sum,  x—y,  is  the  cube  root  required;  for  if  we  cube  x—y  by 
§  63,  we  get  xr'-Sx'y  +  Sxy^—yK 

Example  2.  Find  the  fourth  roots  of  16x^—96x^y  +  21QxY— 
216xV  +  8l2/^ 

This  expression  has  five  terms,  and  the  first  and  last  terms  are 
perfect  fourth  powers.  Taking  the  fourth  roots  of  these  two 
terms,  we  get  ^^320;  and  J^Sz/,  respectively. 

Hence  the  fourth  root  will  be  either  2x^-{-^y^  2x^—3y,  —2x^ 
+  3^,   or  —2x^—3y.     Of  these,   the  fourth  powers   of  2x^—3y  or 
-2x^  +  Sy  will  give  16x«-96^«i/  +  216xY-216xV'  +  8l2/*. 

Hence,  the  fourth  roots  are  2x^—3^  and  —2x^  +  3y. 

From  these  examples  it  is  seen  that 

If  a  perfect  cube  contains  just  four  tenns^  arranged  according 
to  the  poicers  of  some  letter^  the  cube  root  is  the  sum  of  the  cube 
roots  of  its  first  and  last  terms  y  and  if  a  perfect  fourth  poicer 
contains  just  five  terms^  arranged  according  to  the  powers  of 
some  letter^  its  fourth  root  is  the  sum  of  the  fourth  roots  of  its 
first  and  last  terms  ;  and  so  on  for  other  poicers. 

The  terms  of  the  nth  root  of  an  expression  must  always  be 
given  such  signs  that  when  the  root  is  raised  to  the  iith  power 
by  the  method  of  §  63,  the  result  will  be  the  given  expression. 


POWERS  AND  ROOTS  91 

Tlie  work  should  be  checked  by  seeing  if  the  root  will  produce 
the  given  power. 

EXERCISE  26. 

Find  the  cube  roots  of 

1.  x'  +  V2x'  +  4Sx  +  Q'^.  3.  8a;^  +  36a;-  +  54.^  +  27. 

2,  x'-Ux'  +  7bx-12^.  4.  27cf;'-10Sa'b  +  lUab'-Ub\ 

5.  64^«-144e^y  +  108a^y-27/. 
Find  the  fourth  roots  of 

6.  a'-4:'ab^Qa'b'-4ab'  +  b\ 

7.  x'  +  20x'-\  150^^  + 500a; +  625. 

8.  Sla''-4^2a'P'}'SQ4a'b'-lQSc(;'b'  +  2^Qb'\ 
Find  the  fifth  roots  of 

9.  aj5-10a;*  +  40£c-^-80a;^  +  80a;-32. 

10.  S2x'  +  240x\i/  +  720£cy  +  1080£cy  +  810a^y*  +  243y^ 
Find  the  sixth  roots  of 

11.  729a«-1458tr^^^  +  1215«V-540a^5«  I  Uba'b'-Uab''  +  b'\ 

12.  x''  +  12x''a  +  QOx'W  +  160icV  +  240icV  + 192^3^^  _|_  q^^b 


CHAPTER  VIII. 
SPECIAL  PRODUCTS  AND  QUOTIENTS. 

73.  There  are  some  especially  important  products  and  quo- 
tients which  it  is  essential  that  the  student  should  master  before 
proceeding.  They  are  fundamental  forms  that  are  often  met 
in  algebra.  The  student  should  learn  to  write  out  the  pro- 
ducts or  quotients  that  come  under  these  forms  by  using  the 
rules  or  formulae  without  performing  the  actual  multiplications 
and  divisions. 

PRODUCTS. 

74.  Product  of  the  sum  and  the  difference  of  two  terms. 
By  actual  multiplication, 

{a  +  b){a-b)=a'-b\ 

when  the  symbols  a  and  h  stand  for  any  terms  ichatever. 

That  is,  the  p)roduct  of  the  sum  and  the  difference  of  the  same 
two  expressions  equals  the  diff^erence  between  their  squares. 

Example  1.     Find  the  product  of  x-\-2  and  x—2. 
{x  +  2){x-2)=x'-2'' 
=x^-4.. 

Check.     When  x=4:  ;  factors  are  6  and  2  ;  prodnct=12. 

Example  2.     Find  the  value  of  (2^^+  6a')  {2x'-5a'). 
{2x'  +  5a^)  (2x''  -  5a^)  =  {2x'f  -  {^a^ 
=4x*—2oa\ 

Check.     When  x=l,  a=l  ;  factors  are  7  and— 3  ;  product=  —21. 

Example  3.     Find  the  product  otx  +  y  +  5  and  ,r  +  ?/ — 5. 

92 


SPECIAL  PRODUCTS  AND  QUOTIENTS  93 

By  grouping  terms,  these  trinomials  can  be  written  in  the  type 
form  of  the  binomials  a-^h  and  a—h. 

We  have  x-\-y->t-^=  (-r  +  i/)+  5  ;  x-\-y  —  ^=  (x^y)  —  5. 
Hence  ix  +  y  +  5){x  +  y—5)=[(x  +  y)  +  5][(x+y)  —  b] 

=  {x  +  i/Y-5' 

=x'^  +  2xy  +  y^—2o. 

Check.    When  x=l,  y=l\  factors  are  7  and— 3;  product  =—21. 
Example  4.     Write  out  the  product  {a  +  h-\-c)  (a—b—c). 
Grouping  terms,  a-\-b'\-c=a  +  {b-\-c)  ;  a—b—c=a—{b-\-c). 
Hence  {a-\-b  +  c)(a—b—c)  =  {a  +  b-\-c){a—b-\-c). 

=a^-{b-\-cy 

=a'-(jb^  +  2bc  +  c') 

=a^-b''-2bc-c\ 

Check.  Whena=4,  6=2,  c=l;  factors  are  7  and  1;  product  ;=7. 

EXERCISE  27. 

Write  out  the  following  products  without  performing  the 
actual  multiplication : 

1.  {x  +  y){x-y).  ^^  10.  0«^  +  y^)(a3^-y^). 

2.  {m-n){m-Vn).  11.  {a''-b'){d'  +  J)'). 

3.  (a— 5)  (a-f-5).  12.  (m«— /i'')(m*^  +  n*'). 

4.  (a  +  10)(a-10).  13.  {xy-\-ab){xy-ah). 

5.  (y-3)(y  +  3).  14.  (xY-z>){xY-Vz'). 

6.  {a  +  x){a-x).  15.  {'lx'-hy'){^x'-\-by^). 

7.  (2a  +  3)(2a-3).  16.  (3a^-75^)(3a^  +  7^0- 

8.  (3a;-2y)(3a5  +  2y).  17.  (4aic-^  +  5%)(4aar'-5%). 

9.  (5m-4?z)(5m  +  47z).  18.  (\x'^-ly'){\x'-lf). 

19.  (2.5a^-1.7^)(2.5a^  +  1.76). 

20.  {^a'x' -  88iy)  (3|a V  +  33  ly*) . 

21.  (a!H-y  +  2)(a!  +  y-2).  22.  (a^-y-8)(a!-y  +  8). 


94:  ALGEBRA 

23.  (x  +  a  +  b)(x-a-b).  25.  (r'-r  +  l)(r'  +  r-^l). 

24.  f2a;-3y  +  4)(2a;  +  3y-4).      26.  (x-l)(x^l)(x'  +  l). 

27.  (s'  +  s  +  l){s'-s  +  l)(s'-s'-{-l). 

28.  (x*-4:)(x'  +  A)(x'  +  lQ). 

29.  {x  +  y-i-z)(x'{-y—z)(x—i/-{-z)(—x+y+z). 

30.  (10a;"  +  a»)  (a"-10«"). 

31.  (af+'  +  y-')(x-+'-y--'), 

33.    ^-i  +  J-VJi-l-Y 
\4x'     2i/y\4x'      2yV 

75.  Product  of  two  binomials  having  a  common  term  bs  (x  +  a) 

(jr  +  b). 

By  multiplication, 

(x  +  a)(x  +  b)  =x'^  +  ax  +  bx  +  ab. 

Adding  like  terms,  this  becomes  x^-^{a-^b)x-hab. 

Hence,  (x-^a)(x-{-b)  =  x~  +  (a-^b)x-\-ab. 

That  is,  the  j^^'odiict  of  tvno  binomials  having  a  common  term 
equals  the  square  of  the  common  term^  plus  the  product  of  the 
common,  term  atid  the  sum  of  the  other  terms^  plus  the  product  of 
the  other  terms. 

Example  1.     (x'  +  2)(e;i;  +  3)=x2  +  (2  +  3)x  +  2-3 

Check.     When  a?=l  ;  factors  are  3  and  4  ;  product =12. 

Example  2.     {x-4)  {x  +  2)=x'+  (-4  +  2)a?  +  (-4-2). 

=a?'-2x-8. 

Check-     Let  x=l  \  then  (-3-3)  =  l-2-8,  or-9=-9. 


SPECIAL  PRODUCTS  AND  QUOTIENTS  95 

Example  3.  {5x'-{-2y''){5x''-7y')  =  {5a^Y-\-{2y'-7y'')5x'-\-2y'{-7if) 

=2^3C^—2^xhf-l^y\ 

Check.     Let  x=2^  and  2/=3.     (Left  to  the  pupil). 

Example  4.     (a  +  fo  +  5)(a  +  6— 2)  =  (a  +  6  +  5)(a  +  6— 2) 

=  (a  +  6)'  +  (5-2)(tt  +  5)-(5-2) 
=a^  +  2a&  +  62  +  3a  +  36-10. 

Check.     Let  a=l  and  h=2.     (Left  to  the  pupil). 

Note. — In  many  of  the  examples  worked  out  in  this  book  hereafter 
the  process  of  checking  will  he  omitted,  in  order  to  save  space.  .But 
the  student  is  advised  to  always  check  his  work.  The  liabit  of  check- 
ing cultivates  the  indispensable  habit  of  accuracy. 

EXERCISE  28. 

Write  out  the  following  products  ; 

1.  (^  +  3)(^  +  4).  15.  {A^-^){A'-1), 

2.  (a;  +  7)(i»-3).  16.  (2+i>)(jt>-5). 

3.  {h-Q){h-b).  17.  (mV  +  8)(6  +  mW). 
4  (ic-10)(i«  +  2).  18.  (6-a;)(12-a;). 

5.  (a-8)(a  +  6).  19.  (3-a)(10-a). 

6.  (m-ll)(m-2).  20.  (-c  +  5)(-c-7). 

7.  {x-b){x-^).  21.  (a;»  +  3)(£c''  +  7). 

8.  (s-10)(«-3).  22.  (a;" -3) (a;" -5). 

9.  (jt)^  +  12)(jt>^  +  10).  23.  (««+^-6)(«"+^  +  5). 

10.  (a;=^  +  6)(a^^  +  8).  24.  {B'-4.AC){B'-QACy 

11.  (r^-5)(r^-3).  25.  («  +  5  +  3)(a  +  ^>-2). 

12.  (a;='  +  12)(a;2-4).  26.  (aj-y  +  7)(«-y+ 3). 

13.  (a;^H-12)(a;^-3).  27.  (i?  +  ^  +  10)(jt>  +  ^-16). 

14.  («;y^-4)(l+a;2/-^).  28.  (a;^  +  a;  +  6)(a;^  +  a;-3). 


96  ALGEBRA 

29.  (««-««  + 10)(a«-«^-20).      32.  (x'-l  +  x)(x'-2-^x), 

30.  (xy  +  xy  +  xi/)  (xy  +  xy  33.  (x'  -  a')  (x'  -  3a^). 

—  3a;y).  34.  (aj«  +  3y»)(a;"  — Ty").   . 

31.  {z  +  b-b)(z^7-b).  35.  (iC«+^-22/"-^)(3y»-^  +  a;"+0. 

QUOTIENTS. 

Since  division  is  the  inverse  of  multiplication,  from  the  type 
forms  of  multiplication,  certain  type  forms  of  division  follow. 

.76.  Difference  of  two  squares.     Since,  by  §  65, 
(a  +  b)(a—b)=a~  —  b%  then 
a'-b 


a-^b;     _  .  L  =a  —  b- 


fl-6"~""'     a  +  b 

That  is,  the  quotient  obtained  by  dimding  the  difference  be- 
tween the  squares  of  tioo  expressio?is  by  the  diff^'erence  between 
those  expressions  equals  the  su7n  of  the  expressio9is.  And  the 
quotient  obtained  by  dividing  the  difference  betioeen  the  squares 
of  two  expressions  by  the  su7n  of  those  expressio7is  equals  the 
difference  between  the  expressions. 

EXAMPLE  1.  ^^  =  (£!rr:(5)^=^.  +  5. 

XT—    5        X*— 5 

Check.  Whena?=l:  divisor=  — 4;  dividend  =—24;  quotient=6. 
E^^^^^^-  2-     9a^a^  +  2y^  ^  9a-.^+2^-       "^^  ^^"^^  • 


EXERCISE   29. 

Perform  the  indicated  divisions  : 

1    x'-9  o    «*— 25  K    4ic' 


3  'a'-  5  2x-l 


a; +3*  '     ^=^  +  4  3a +1* 


1 

—  lax 

1- 

-IQx^ 

1- 

-4.x*  ' 

r- 

-1 

SPECIAL  PRODUCTS  AND  QUOTIENTS  97 

y    2^a'^l        ..    64x*-Sly'  2ba'-22^b' 

'     ^ab+1   '  '     Sx'-Qf    '  ^^'      ba-lbb'   ' 

'     ^      "        •  lla^«+10a    *  a«-6«* 

1  —  Kymrw 
10    ^'"-^  14    169a;y-144a^  -       («  +  ^)2_^2 

•  1  +  ^^*  ■       V6xy'^Vla'    '  -^^5-— • 

a  +  h—x—y     '  '    {a  +  by-\-{x—yf 

77.  Sum  and  difference  of  like  powers  of  two  expressions. 
By  actual  division, 

^■^—^  =  a'  +  ab  +  b'; 
a  —0  ' 

^lz^'  =  c(?-\-ceb-Va¥-VW', 
CI/      0 

d'^  —  b^  is  not  divisible  by  a  +  J; 

^^J^ = d' -  cr5  +  «6^  -  h' ; 

d^^-b^  is  wo^  diinsible  by  a— ^  ; 

a*  +  ^*  is  7/^0^  diuisible  by  a—b\ 

d^  +  b^       2        7  I  Z.9 
z=a'  —  ab  +  b^\ 


a  -{-b 

a^  +  5Ms  ?io^  divisible  by  6«  +  5. 

These  examples  illustrate  the  following  principles.* 
(a)  a"  —  b*  is  always  divisible  by  a  —  b. 

The  quotient  is  a''-^  +  a''-264-a"~^5^+ +6"~^ 

Thus,  a'^—b^,  a^—¥,  a*—b*,  etc.,  are  divisible  by  a—b. 

*  The  general  proof  of  the  divisibility  in  these  various  cases  is  left 
for  the  student  in  Exercise  43. 
7 


98  ALGEBRA 

(b)  a"  —  b"  is  divisible  by  a-\-b  only  'when  n  is  even. 

The  quotient  is  a"-^  —  W'-'^b  +  a''-W— —  6«-i. 

Thus,  a^—b^,  a*—b\  a^—lf^  etc.,  are  divisible  by  a  +  6  ;  whil^ 
a^— 6^*,  a^—b^,a'—b\  etc.,  are  not  divisible  by  a  +  b. 

(c)  a''-^b''  is  never  divisible  by  a  —  b. 

Thus,  a^  +  6^  a^  +  6%  a*  +  b\  are  not  divisible  by  a—b. 

(d)  fl"  +  6"  is  divisible  by  a^b  only  when  n  is  odd. 

The  quotient  is  a"-'^—a>'-^b  +  a''-W— +6""^ 

Thus,  a^4-6^  a^  +  6^,  a^  +  6^  etc.,  are  divisible  by  a  +  &,  while 
a^  +  b^,  a*  +  b\  a^  +  b^^  are  not  divisible  by  a  +  b. 

From  the  preceding,  observe  that  : 

W7ien  the  divisor  is  a  —  b,  the  signs  of  all  terms  of  the  quo- 
tient are  +. 

When  the  divisor  is  a  +  6,  the  signs  of  the  terms  of  the  quo- 
tient are  alternately  -\-  and  —. 

The  exponent  of  a  in  the  first  term  of  the  quotient  is  less  by  1 
than  the  exponent  of  a  in  the  dividend.,  and  decreases  by  1  in  the 
successive  terms. 

The  exponent  of  b  is  1  in  the  second  term  of  the  quotient .,  and 
increases  by  1  in  the  successive  terms. 

Example  1.     Divide  a^—Whj  a—b. 

^^  =a'  +  a'b  +  a'¥  +  a'b'-\-ab*  +  b\ 

Check.  When  a=2,  b=l  ;  dividend=63  ;  divisor=l  ;  quo- 
tient =63. 

Example  2.     Divide  64^2/'  +  i25  by  4xy  +  5. 

Uoc^y^  + 125  ^(4xyf +  5^ 
4:Xy-^5     ~   Axy  +  5 

=  (4xyy-(4:xy)  (5) +  5' 
=^lQx'y'-20xy  +  25. 


SPECIAL  PRODUCTS  AND  QUOTIENTS  99 

Example  3.     Divide  32a^-2436^"  by  2a-Sb\ 

32a^  -  2436^  ^  (2af  -  {Sb'f 
2a-3b'  2a-W 

=  {2ay  +  (2a)^(36*0  +  {2ay{^by  +  {2a)  {Wf  +  {W)' 
=  16a*  +  24a='  h'  +  36a'  b'  +  54a6«  +  816«. 

EXERCISE  30. 

1.  Can  a*— 1  be  divided  by  a-\-l  ? 

2.  Can  cc'—af  be  divided  by  a—x? 
3-  Can  a'—x^  be  divided  hj  a  +  x? 
4.  Can  a'  +  i'  be  divided  by  a  +  ^*  ? 

By  Avhat  expressions  can  the  following  be  divided  ? 

6.  ay'  +  S.       6.  l-£c\         7.  a;^-32.       8.  a=  +  ^>l       9.  xy-l. 

Determine  which  of  the  following  indicated  divisions  are 
possible,  and  write  out  all  possible  quotients  : 


10. 

m—n 

11. 

1-/ 
l+y- 

12. 

a^  +  1 
a-1* 

13. 

aM-1 
a  +  1* 

14. 

x'-l 
x-1' 

15. 

a'-b' 

IB 

16a«-81J* 

18. 
19. 
20. 
21. 


x'^'y" 
x^-\-y^ 

'xTy" 

a*  +  b* 
a—b' 


y 


x-y 

22.^' 

x+y 

23.^ 


'Id'  +  Zb'  xy'-a'  «•  +  ! 


100  ALGEBRA 

EXERCISES  FOR  REVIEW  (II). 

1.  What  is  the  rule  for  adding  similar  terms  ?  From  what 
law  does  it  come?  Add  ^xihj^  —\Qi'}y^  |a?^y,  —  6a;-y,  —\xhj. 

2.  How  do  you  add  dissimilar  terms  ?  Add  2a^,  —  3a;,  —  2c, 
^x\ 

3.  Simplify  3.T^- 22/ -7y  +  6£c^  + 2«. 

4.  In  what  letter  are  the  terms  3icy^,  2aa?,  Zcxy  similar  ?  Add 
them. 

6.  How  do  you  add  polynomials  ?  Add  6a;^— 2£c  +  5,  —^x^^ 
4i«-l,  7a;^-5£c-2. 

6.  What  is  meant  by  checking  work  in  algebra  ?  How  would 
you  check  the  result  of  exercise  5  ?     Check  the  work. 

7.  Add  d^—y^^  ^a^y—xy'^^  x^  +  ^xy'^—y^.     Check  the  work. 

8.  How  do  you  subtract  polynomials  ?  From  3c«"'  — 2«^  +  a— 4 
take  a?  +  4a^  + 1.     Check  the  result. 

9.  What  are  the  laws  of  signs  to  be  observed  in  removing 
signs  of  grouping  ? 

10.  What  is  indicated  by  6£c-4y  — (3a?— 2y)  +  (5£c  +  ?/)? 
From  what  fundamental  processes  do  the  laws  of  grouping 
follow? 

11.  Simplify  a-[3^>+ {3c-(c-5)  +  «} -2a]. 

12.  What  laws  must  be  observed  when  inserting  signs  of 
grouping  ? 

13.  Group  like  terms  in  x  so  as  to  have  the  sign  +  before 
each  sign  of  grouping :  Ix^  —  3c^£c — ax^  +  5ic  +  Ix^ — ahx". 

14.  Group  like  terms  in  x  so  as  to  have  the  sign— before  each 
sign  of  grouping  :  lyx — ax^  —  hx"  +  3a;^ — ^ax — ^bx?  +  2ic^ — ex?. 

15.  Add  by  combining  like  powers  of  x\  «^— 2a;,  aa;^  +  5, 
ax'-Zx^^-hx^^ 


EXERCISES  FOR  REViEW  101 

16.  What  is  the  law  of  exponents  in  multiplication  ?  Find 
the  value  of  a^a^ ;  a-a}^-(]^\  x^-x'-x^. 

17.  How  do  you  multiply  monomials  ?  Find  the  product  of 
—  2£cy,  Za^xif^  —ax^,  and  —  |^.y. 

18.  What  is  the  meaning  of  x'  ?     Of  x"  ?    Of  (xy  ? 

19.  From  what  law  do  we  obtain  the  rule  for  multiplying  a 
polynomial  by  a  monomial?  State  the  rule.  Multiply 
x'-'2x'  +  Sx-bhj  2x\ 

20.  How  can  you  check  your  work  in  multiplication? 
Check  the  work  in  the  preceding  multiplication. 

21.  From  what  do  we  obtain  the  rule  for  multiplying  a  poly- 
nomial by  a  polynomial?  Multiply  2a^^  — 3a''  +  26'  by  d^~ab 
-\-b\  and  check  the  work. 

22.  Simplify  ^[^ab-'2a{b-4(a-b)}']. 

23.  What  is  the  law  of  exponents  in  division  ?  Find  the 
values  of  «'"^-a^ ;  a^-^a^ ;  a^~a\  a}-^a^. 

24.  What  is  the  meaning  of  a^  ?     How  is  it  shown  ? 

25.  How  do  you  divide  a  polynomial  by  a  monomial  ? 
Divide  a'-'2a;'b'  +  b'  by  a'-2ab  +  b\ 

26.  What  is  the  relation  between  the  dividend,  divisor,  quo- 
tient and  remainder  ? 

27.  What  is  a  fraction  ?  - 

X 

28.  Prove  that  ~y=x. 

29.  How  do  you  find  the  product  of  two  or  more  fractions  ? 
TV  ^  ^^  ^ 

y    21a;    1 

30.  {ay='i   («»)"•=?   {by{my='i   (3^)*=?     State  the  law. 

31.  {xyy='i  {xYy=?  {4d'b'y=?  State  the  law  that  you 
used. 


102  ALGEBRA 

32.  What  are  the  laws  of  signs  in  involution?  {  —  2x^yh^y^  =  '> 


%7      • 

33.  What  is  the  square  of  a-\-b?  Show  that  your  result 
gives  a  rule  for  squaring  any  binomial.     (2x^  —  Sy*y-=? 

{la'  +  Shy  =  ?  {x'-2y=? 

34.  Square  «  + ^  +  c  and  show  that  the  result  gives  a  rule 
for  squaring  trinomials.  Square  ^x—Sy  +  zhy  the  formula  you 
have  just  derived. 

35.  What  is  a  root  of  a  number  ?     What  is  the  index  f 

36.  What  are  the  laws  of  signs  in  evolution?  ]/4iK^=? 
f/27a^^-?  f'-32a'"^>'^=?   fl^hf=^.   |/'=4=? 

37.  What  is  a  perfect  nth  power  ?    Illustrate. 


38.  i/16£c*-8£c^a  +  a^  =  ?  |/81  +  25/iM-90??;^  =  ? 
f/8a;'^-36ajy  +  54£cy-27y'*  =  ? 

39.  Find  the  product  0,1  a— h  and  a-\-h  and  show  that  this 
gives  a  rule  for  finding  the  product  of  two  binomials. 
(4£c'  +  3y-)(4a;^-32/=^)-=?  (i-6cc-^)fi  +  6£c^)-? 

40.  Find  the  product  of  a^  +  a  and  x^h  and  show  this  gives 
a  rule  for  finding  the  product  of  two  binomials  which  have  a 
common  term.    (a;^  +  6)(a;^-4)  =?  i^lah  +  3)(2a^>  +  7)  =? 

(4c.^-3)(4c£c  +  ll)=? 

41 


^'-y'_^          ^.B_y6_^          a^«+y«_, 

X  —y      '            x^-y^     '            sc^+y' 

d'  +  l 

42.  AVhen  is  a"  —  b"  divisible  by  a  +  b? 

43.  When  is  a'^  +  b"  divisible  by  a  +  6? 

CHAPTER  IX. 

FACTORS. 

Definitions  and  type  forms. 

78.  Factors  were  defined  in  §  12.  It  follows  from  the  prin- 
ciple quotient  X  diinsor  =  dimdend^  that  a  factor  of  an  expres- 
sion is  an  exact  divisor  of  it.  The  process  of  obtaining  the 
factors  of  a  given  expression  is  called  factoring.  Hence  factor- 
ing, like  division,  is  the  inverse  of  multiplication  and  depends 
upon  certain  type  forms  established  by  multiplication. 

Thus,  since  (a  +  6)  (a— 6)=a^— 6^  the  factors  of  a"-'— 6''  are  a-\-h 
and  a—h. 

A  common  factor  of  two  or  more  expressions  is  an  exact  dioi- 
sor  of  each  of  those  expressions. 
Thus,  a  is  a  common  factor  of  a&,  ac,  and  ax+ay. 

It  is  understood  that  in  this  chapter  only  rational  factors 
of  an  expression  will  be  considered. 

79.  Any  monomial  expression  can  be  factored. 

Thus,  6x^y^=2Sxxyyy.  Hence  its  factors  are  2,  3,  x,  x, 
2/,  2/,  y- 

80.  Not  all  polynomials  can  be  factored  into  rational  factors. 
But  there  are  certain  types  of  polynomials  which  can  be  factored. 
These  types  will  be  discussed  in  this  chapter. 

8 1 .  Monomial  factors.  A  polynomial  containing  a  monomial 
factor  may  be  factored  by  aid  of  the  distributive  law : 

ffjr+6jr+cjr+ =(a  +  6  +  c+ )x 

103 


104  ALGEBRA 

This  identity  shows  that  x,  which  is  a  factor  of  ever  i/  term  of 
the  polytiomial  ax  +  bx  +  cx-\-   •  •  •  •  ,  is  a  factor  of  the  poly- 

7iomial  itself     And  the  other  factor^  «  +  6  +  c+ ^may 

he  obtained  by  dimding  the  given  polynomial  by  the  monomial 
factor. 

Hence  the  rule : 

Find.,  by  inspection,,  a  monomial  vihich  ivill  divide  every  term 
of  the  polynomial.  Divide  the  given p)olynomial  by  this  mono- 
mial. The  divisor  and  quotient  are  the  monomial  and  polyno- 
mial factors.,  respectively.,  of  the  given  jyolynomial. 

Example  1 .     Factor  2ax'^ —4ay'^  +  6az\ 

By  inspection,  2a  is  seen  to  be  a  factor  of  each  term.  Hence  2a 
is  the  monomial  factor.  Dividing  by  2a,  we  get  x'^—2y^-\-oz^,  the 
polynomial  factor. 

The  factor  2a  may  itself  be  factored  into  2  and  a.  Therefore 
all  of  the  factors  of  2ax^  — 'iay^  +  ^az^  are  2,  a,  and  x^—2y^-{-2>z^. 

The  given  polynomial  may  be  written  2a{x^—2y^  +  Sz^). 

Note. — The  factors  2,  a,  and  x^  —  2y^+^z'^,  no  one  of  which  can  be 
factored,  are  sometimes  called  the  prime  factors.  2a  is  called  a 
composite  factor. 

Example  2.     Factor  4:xy^—2x'^y^+x^y. 

Each  term  may  be  divided  by  xy.  Dividing  by  xy^  gives 
4y^—2xy-{-x\ 

Hence  the  required  factors  are  a?,  y^  and  4:y^—2xy  +  x^  ;  and 
4xy^ — 2x'^y^  -\-  x^y  =  xy{'iy'^ — 2xy  -f-  ^0- 


EXERCISE  31. 

Factor : 

1.  xy'^—xy-\-x. 

5.  xY-\-xY- 

2.  a5='y  +  3£cy-5a;y^ 

6.  bd'-l^a'b. 

3.  a;'— 3a;. 

7.  2^x'  +  lxY, 

4.  a;'  +  5a;^ 

8.  18a;^-9aJ^ 

FACTORS  X05 

9.  a''-^cv>h  +  2a*b\  15.  24a;y-12ar'y  +  42a!y. 

0^,  6a;*  +  9icy  +  3a;y  ^6.  27wVi  +  36mW  +  81wi/il 

11.  'Ua'b''^ZMb\  ^7.  56ay-14ay +  28ay. 

12.  4x'  +  4:x\  18.  ic"  +  a£c«. 
^3.  8a'*  +  4a^^>  +  2a«^>^                    l9.  a^?r'-a\ 

14.  «^^>V4-«'^>'c'''  +  «'^^V.  />^0.  5a«y«-^  +  10a"-V. 

82.  Polynomials  that  are  powers  of  binomials.  Polynomials 
that  are  poioers  of  binomials  may  be  factored  by  aid  of  §  71  and 
§  72.  In  any  polynomial  the  monomial  factor,  if  one  exists, 
should  first  be  discovered  and  divided  out. 

Example  1.     Factor  a?*— 2ay*2/  +  ^*V- 

By  inspection  x'^  is  seen  to  be  a  mono7nial  factor.     Dividing  by 
a?^,  we  get  x^—2xy  +  y'^.     This  is  the  square  oix—y. 
Hence,  x*—23c'y  +  x'^y'^=x^{x'^—2xy-\-y'^) 

=x~{x—yY. 
The  factors  are  a?,  x^  x—y^  ^c—y. 

Check.     Whena?=2,  i/.=l;  polynomial=4;  factors  are  2,  2, 1, 1. 

Example  2.     Factor  ^^xHj-2Ux^y'  +  ^2^x'y^-1^2xyK 

Qxy  is  a  monomial  factor.     Dividing  by  Qxy^  we  have 
8x^  —  3Qx^y  +  54:Xy^—27y^.     This  quotient  is  the  cube  of  2x—3y. 

Hence,  /"T^ 

.    4Sx*y—21Qafy''  +  S24x^y''-162xy*=6xy{8x'-(^^y  +  Mxii^-27y")  • 

=6xy{2x-3tjf.    ' 

The  factors  are  6,  a?,  ?/,  2x—Sy,  2x—3y,  2x—3y. 

Check,  When  x=l,  y=^\  polynomial  =—768;  factors  are  6,  1, 
2,  -4,  -4,  -4. 

Example  3.     Factor  —x^  +  2xy^ — y*. 

Taking  out  the  factor  —1,  we  have  a  perfect  square. 

Hence,  —x'  +  2xy'-y*=-l{x^-2xy^-\-y*) 

=  -l{x-yy 

=  -{x-yy. 


106  ALGEBRA 

EXERCISE  32. 

Factor : 

1.  bx'-40x  +  S0.  My.  -a'  +  Sa'-16. 

2.  4:a'-Sab  +  4:b\  8.  2ba'b'-10ab'x'y-{-b'xY. 
)^3.  ax'  +  Qax'  +  9a.                      \9.  Sx'(/-Sxy  +  ^xf. 

4.  4x'f-^Sbxij  +  4b\  10.  2SSa'  +  4S0a'b  +  200b\ 

J>(p.  20a'-60a'  +  4ba.  >L1.  7x'  +  21xh/  +  21xi/'  +  7i/\ 

6.  Qx-9x'-l.  Vl2.  ba'-lbd'b  +  lba'b'-^ab\ 

M^.  2x'^-{-QxY  +  QxY  +  2xy\ 
>(i4.   -^a'x-\-4:^a'bx-lSba'b'x+U^a¥x. 
VI 5.  32a* -  64a=^^  +  48a^^>^ -  IQab'  +  2/A 
Vl6.  Sa'-na'b  +  SOa*b'-nOaW  +  na'b'-Sab\ 

8  3 .  Trinomials  of  the  form  x'-{^ax  +  b.      By  §  75, 
{x  +  m){x-i-n)  =  x^  +  (m-\-n)x-\-mn. 

Now  a;^  +  (^  +  ^?')£c  +  m/i  is  of  the  form  x'^  +  ax  +  b,  where 
m-\-7i  =  a  and  7n?i  =  b. 

But  the  factors  of  x'^-{- (m  +  n)x  +  mn  are  cc  +  m  and  x  +  ii. 
Hence,  if  x^-hax  +  b  has  rational  factors,  they  vnll  consist  of 
two  binomials,  like  a^  +  m  andx-\-n,  having  the  common  term  x, 
and  the  other  terms  such  that  their  sum  is  a  and iwoduct  b. 

Thus,  since  (a? +  3)  (a?  +  4)=x'  +  7a7+12,  the  factors  oix^-\-lx-\-\2 
are  a?  +  3  and  x-\-4.     Here  7=3  +  4,  and  12=3x4. 

Example  1 .     Factor  qc}^^x^  18 . 

Here  the  factors  must  have  the  common  term  x,  and  the  other 
terms  of  the  two  factors  must  be  such  that  their  sum  is  9  and 
product  18.  Two  numbers  whose  su7n  is  9  and  product  18  are 
3  and  6. 

Hence     x''  +  9x  +  18={x  +  3)(x  +  Q). 

Example  2.     Factor  a?2—2x— 35. 


FACTORS  107 

Evidently,  here  we  seek  two  numbers  whose  product  is  a  nega- 
tive number^  —35,  hence  the  numbers  must  have  unlike  signs. 
And  since  their  sum  is  —2,  the  one  having  the  greater  absolute 
value  must  be  negative.     Hence,  they  are  5  and  —7. 

Therefore,  x^—2x—i^5  =  {x—7){x+5). 

Example  3.    Factor  f -25^  + 150. 

The  common  term  here  is  t.  Since  the  product  of  the  other 
two  terms  is  +150,  they  must  have  like  signs.  And  since  their 
sum  is  —25,  they  must  both  be  negative.  Hence  these  terms 
are  —10  and  —15. 

Therefore,  t'-25t  +  loO=(t-10){t-15). 

Example  4.     Factor  3—x^—2x. 

This  can  be  thrown  into  the  type  form,  x'^-\-ax+b,  by  taking 
out  the  factor  —1. 

Then  3-x'-2x=-l  {x'  +  2x-S). 

But  x''  +  2x—3=(x+S)(x-l). 

Hence,  3—x'—2x=  —  lix-\-S){x—l), 

or  multiplying  first  and  last  factors,  this  may  be  written 

{x  +  S)(l-x). 

Example  5 .     Factor  a'^x*  +  5a V  +  6 . 

This  is  of  the  form  x^  +  ax-\-b,  which  may  be  more  easily  seen 
by  writing  a*x^  +  5a^x^  +  6  =  (a'^x'^y  +  5  (a V)  +  6 . 

The  X  of  the  standard  form  is  a^x^  in  this  exercise,  hence  the 
common  term  of  the  factors  will  be  a^x"^. 

Therefore,  {a'xy  +  5 (a^x^)  +  6  =  {a'x''  +  3)  (a V  +  2) . 

Example  6,     Factor  x^  +  ^xy  + 14^/^ 

If  we  write  this  x^  +  {^y)x-\-lAy'^,  it  is  seen  to  be  of  the  given 
type  form. 

The  common  term  is  x  and  we  have  now  to  find  two  expressions 
whose  sum   is  9^/  and  whose  product  i^  142/^. 

It  is  easily  seen  that  these  two  expressions  are  2y  and  7y. 

Hence,  x^->r^xy-irl4.y^={y-\-2y){x  +  7y). 


108 


ALGEBRA 


Factor : 

1.  x'  +  lSx  +  42.  8. 

2.  x'  +  2x-4S.  9. 

3.  £c^-9a^  +  20.  >\^0. 

4.  x'-Sx-2S.  11. 

5.  a;^  +  17^  +  72.  ^12. 

6.  x'  +  bx-bO.  13. 

/,.-^2.  x'-^bx'  +  Q. 
23.  aj*-7£c^  +  10. 

26.  fc«  +  3£cM-2. 

27.  a^^'-Sa^^+e. 

28.  a;^«  +  lla;-'-26. 

29.  a;'^-2a;«-224. 

30.  a'x'  +  9ax+-U, 

31.  bhf-lhy-2,0. 

32.  a^^^  +  30a^>  +  200. 

33.  £cy-28a!y  +  160. 

34.  mV  +  4m?i-60. 
Y35.  «^^V  +  13r^^6— 30. 
^36.  a;ys^-19£cy^  +  90. 

37.  xy-^xY  +  2. 

38.  a;y  +  14£cy  +  33. 

39.  a;y-5ajy-126. 
^40.  «'"^>^-2«-^^>-35. 


EXERCISE  33. 

a;^-12a;  +  32.  15.  c^417c-84. 

a^  +  3a-180.  ^\l6.  cP-M-bb. 

m'-m- 240.  17.  2-{-Sr  +  r\ 

f-t-420.  M8.  24:-2s-s\ 

a2+3«  +  ^.  19.  15  +  2y-yl 


3c-70. 


^JO. 


2-rt-a^ 


5'''  +  20Z>+84.         21.  z'-z'-2. 

41    a;y  +  7£cy^— 44. 
]s>^42.  pY-Sp'q'  +  2. 

43.  a;=^  +  3a!y  +  2yl 
*      44.  £c^  — 5^cy  +  6y^ 
^5.  a3^  +  17a;y  +  70y^ 
^"^6.  a'^10ab-S9b\ 

47.  a'-lSab-40b\ 
\s,  -^a^-\-2ab  +  b\ 
i)*9.   -«^-5a^»  + 104^1 

lO.  a^y  —  babxy  +  6a^^^ 


>/. 


^61.  xY^^Uibx}f-V^ceb\ 
62.  £cy-3c.^?/-10cl 
53.  a!y  +  9axy  +  14al 


\ 


64.  x^if  —  la^bx^y^  -f  1 2«:«'^6l 

65.  c«^m^  +  llac^w'  +  30c^ 
C^6.  l-3a  +  2a^ 

57.  \^-Qx-21x\ 

58.  (a  +  ^)^  +  7(a  +  ^»)  +  10. 
V59.  {x-yy-^x-y)-40. 


FACTORS  109 

60.  (x  +  yy  +  U(x+>/y  +  SS.       66.  2£c^-34a;-400. 

61.  (a  +  by-S(a  +  b)(x+i/)+   ,66.  ax'  +  bax-Ua. 
2{x-{-yy.  67.  a'x'  +  2a''x-Sba\ 

62.  2ic^-10x-168.  68.  x'  +  ax'-42a'x. 
(First  remove  tlie  monomial  factor. )  69.  dx^  —  QOx^i/  —  288£cy^ 

63.  Sx'^Sx-lS.  ^70.  xy  +  9xy  +  UxY. 

64.  5ic^  +  45ic+100.  71.  260a;  +  62«V  +  2a;y. 

72.  220-2a;-2ajl 

84.  Trinomials  of  the  type  form  ax'^-\-bx+c. 

There  are  different  methods  of  factoring  the  general  quad- 
ratic trinomial  of  the  type  form  ax'^  +  bx-\-c  when  it  has  rational 
factors.  Such  a  form  may  be  factored  by  first  changing  it  to 
the  form  discussed  in  §  83.  Thus,  multiplying  by  a,  and  at  the 
same  time  indicating  the  division  by  a,  in  order  not  to  change 
the  value  of  the  expression,  we  have 


ax^^bx  +c= 


aV- 


abx 


a 

(axy  +  b(ax)  +  ac 
~  a 

This  numerator  is  now  a  quadratic  trinomial  in  («.x'),  the  form, 
discussed  in  §  83  v^hen  x  is  replaced  by  ax.  Hence,  we  have  the 
common  ter7n  ax^  and  the  other  terms  such  that  their  product 
equals  ac  and  their  sum  equals  b ;  then  finally  dividing  by  a, 
we  get  the  required  factors  of  ax^-{-bx-\-c. 

Example  1.  Factor  2xH8a?+l. 

4x^  +  6^^  +  2 


2ic-^  +  3x+l     = 


2 

{2xf  +  2>{2x)-^2 

2 
(2.r  +  2)(2a!+l) 

2 


110  ALGEBRA 

Dividing  the  first  factor  of  the  numerator  by  2,  we  have  (x  +  1] 
(2ic+l). 

Check.  When  x=l  ;  trinomial=6;  factors  are  2  and  3. 
Example.  2.     Factor  Qx^  +  llx  +  S. 

6 

_i6x  +  9)iQx  +  2) 

=  {2x  +  3)(3x  +  l). 

Here,  to  divide  the  product  of  the  two  factors  by  6,  divide  the 
first  one  by  3  and  the  second  by  2. 
Example  3.  Fsictor  ax'^+{a-\-b)x  +  b. 

a 

=  (ax  +  a){ax  +  b) 

a 
=(a?  +  l)(ax  +  6). 

Example  4.  Factor  1 2a?'— 23a?// +  10?/^ 

1/* 

_{12x-8y){nx-15y) 
12 

=  {3x—2y){4x-5y). 

A  second  method  may  be  obtained  as  follows : 

Since  the  factors  are  buiomlals  of  the  form  (mx  +  ri)  and 
{rx  \-s)  whose  product  is  rnix^^{rn  +  sm)x  +  s7i^ 
therefore,  ax^  -\-hx  +  c= rmx^  +  {rn  +  sm)x  -[-  sn  =  (mx  +  ?i)  (rx  +  s). 

From  this  it  is  observed  that  if  a  general  trinomial  of  this 
form  can  be  factored,  the  ^ first  and  last  terms  of  the  trinomial 
must  he  so  factored  as  to  give  the  terms  of  the  hinomial  factors^ 
and  the  sum  of  the  products  of  the  first  term  of  each  hinomial 
factor  by  the  second  term  of  the  other  factor  must  give  the  middle 
term  of  the  trinomiO/h 


FACTORS  111 

Example  1.     Factor  2x^  +  5x+2. 

Now  the  first  terms  of  the  binomial  factors  are  factors  of  2x^, 
and  the  second  terms  are  factors  of  2.  The  sum  of  the  products 
of  the  first  term  of  each  by  the  second  term  of  the  other,  called 
cross-products,  is  ^x. 

The  possible  pairs  of  factors  may  be  conveniently  arranged  as 
follows : 

2x+2.  2a?+l 

x+1  _  x^-2 

from  which  we  may  easily  select  the  pair  that  gives  the  proper 
cross-product.     The  factors  then  are  2a? +1  and  x-\-2. 

Example  2.     Factor  3ir^  +  £c— 10. 

Four  of  the  possible  pairs  of  factors  are 

x—2        x  +  2        x—5        x-\-5 
3x  +  5      3j?— 5       3x+2       3a;— 2 

from  which  it  is  seen  that  the  second  set  is  the  correct  one. 

Each  set  should  he  tested  as  vnritten.  If  this  is  done.,  it  will 
usually  be  unnecessary  to  write  all  possible  sets.  The  simpler  ex- 
pressions of  this  type  can  be  factored  by  inspection. 

Note. — The  student  should  always  look  for  monomial  factors  first, 
and  remove  any  that  are  found.     iTse  the  method  that  seems  best. 


EXERCISE  34. 

Factor : 

1.  2£c^  +  5a;  +  2.  7.  12x'  +  bx-2. 

]2<  Sx'  +  7x-\-2.  8.  6x'-llx  +  b. 

3.  2x'-h9x-\-10.  9.  2t'-bt-l. 

4.  2x'  +  bx-S.  10.  8a;'  +  15a;-2. 

5.  2x'-^x  +  ^.  11.  12c^-25c+12. 

6.  bx'-9x-2.  12.  bb'-79b-U. 


112 


ALGEBRA 


13. 

15. 
16. 

Vl7. 

tA  21. 

23. 

fv    24. 
26. 


7«-  +  36a  +  5. 
10/  +  21y-27. 


2r* 


6. 


Ua'  +  ba'-l. 
6aV  +  13aV  +  6. 
5a'^>V  +  19a^>c— 4. 
2icy — x^i/ — 15. 
Qa'b'-2Sa'b'  +  20. 
12x'  +  21x-Q. 
Qax'^  +  lbax  +  9a. 
8Qx'i/  +  U2xj/-Ui/. 
Qa'b-4:Qab-72b. 
2x'  +  Sxi/  +  y\ 


26.  12x'-7xy+f. 
S  27.  2x'-Sxi/-2i/\ 
K28.  bx'-19xi/-'^tf. 
\29.  12ni'-lQa77i-Sa\ 

30.  2£cy  +  «y-15. 
'Nl.  3a;^  +  10a;y-8/. 
V    32.  21ax^—hlaxy^^ay^. 
•     33.  6a^^»^  +  2«Z>c—4cl 

34.  ax^-\-{b-a)x-b. 
(X35.  2y^  +  (4a  +  %  +  2«^. 

36.  2;2^-(2a  +  ^)2;  +  a*. 
'  37.  ax'-^{ab^-\)x-^b. 

38.  2a£c'''+(2a^-2«^»)a;-2«^^>. 


85.  Binomials  of  the  type  form  a^  —  b^.     Binomials  of  the  type 
form  a^—V^  may  be  factored  by  aid  of  §  74.     Since 

(a  +  6)(fl-6)-a^-6% 

the  factors  of  a^  —  b-  are  a-^b  and  a—b. 

Example  1.     Factor  4x^—25. 

=  (2ic+5)(2a!-5). 
Note. — We  might  also  write 

=  (-2a^-5)(-2if+5). 

Usually,  however,  we  have  no  use  for  this  second  set  of  factors. 

It  is  easily  seen  in  general  tiiat  if  an  expression  has  two  factors,  tlie 
expressions  obtained  by  changing  the  signs  of  those  factors  will  also  be 
factors.     Thus,  the  factors  of  ah  are  a  and  6,  or  — a  and  — 6. 


FACTORS  113 

Example  2.     Factor  27a?^— 12a?. 

Kemoving  the  monomial  factor  3a;,leaves  9a?^— 4. 

=  (3x-2)(3.r  +  2). 
Hence  27x'-12^=3x'(3ic-2)(3x  +  2). 

Note. — We  need  not  write  down  the  second  step  of  the  above  solu- 
tion, but  write  merely 

9^2-4=(3j;-2)(3a;+~2). 


' 

EXERCISE  35. 

Factor : 
1,  x'-m. 

8. 

x^-n\                ^^62bx'- 

-2256^ 

2.  x'-l^jd. 

9. 

4x'—a\                   16.  9a V - 

1. 

3.  0^—49. 

4.  x'-U. 

10. 
11. 

9a3^-a^             17.  i6a^y- 

-9. 

■25. 

5.  a^^-144. 

6.  03^-196. 

12. 
13. 

lQx'-2ba\            19.  25£cV- 
lOOy^-495^      "^20.  lOOx'y 

-16. 

^-815^, 

7.  x'-a\ 

14. 

Sla'  —  64b\            21.  IG^cV- 

-b'c\ 

^  22.  imxYz'-^ha'h 

y%Z.  \x'-\y\ 
24.    1  a2_  1  52 

^cl 

36a.V^      -, 
2^^-    25a^5-^      1- 

33                        1  fiu2 

2  5              16 

K26.  T-V^«'6'-^V«^. 

„.      4x'      81?/ 
V^-    Sly^      4x^ ' 

27.  /T-IKy'- 

\28.  1— Vra'*'«'- 

35.  1          ^,      . 

"  29.  Ti^-T-v*y. 

36.  (a  +  by-1. 

30. 2-:_«i 

„,       a'       25(8' 

^7,  {a+by-(c+dy. 

38.  (x  +  yy-(a  +  by. 
.     39.  (a^-y)'^-(«-^')^ 

114  ALGEBRA 

40.  4(x+(/y-9{a  +  by.  -o     (a  +  by     (c  +  dy 

{a-by   {c-dy 
(2a+by  2Hx+j/y_w(x-j/y 

^  ib{x     Zy).  to.    ^Q^^^j^^y     8i(^^_^)2- 

86.  Polynomials  which  can  be  written  in  the  type  form  a^—b^. 

Some  polynomials,  by  grouping  terms,  can  be  put  into  the 
tjpe  form  a^—b^^  and  can  then  be  factored  by  the  method  of 

■§  85.'     .• 

Example  1.     Factor  aH2a&  + 6='- c^ 
Grouping  terms, 

o?  +  2a6  +  6'  -  & = {a'  +  'itah  +  h^)  -  & 

=(a  +  6  +  c)(a  +  6— c). 
Example  2.     Factor  a^—Jy"— 2bc — c\ 
Grouping  terms, 

=a^-{b  +  cy 

=  {a  +  (b  +  c)}  {a-{b  +  c)} 

=  {a  +  b  +  c)  (a—b—c). 

Learn  to  omit  the  second  and  third  steps  and  to  lorite  out  the 
icork  as  follows  : 

Example  3.     Factor  x"  +  6x— a'  +  4a&— 46'  +  9. 
Grouping  terms, 

x^*  +  6a?  -  a' +  4a6 — 46H  9 = (x^  +  6.r  +  9)  -  (a' -  4a6  +  4&') 
=  (a;  +  3  +  a-26)(a74-3-a  +  26). 

EXERCISE  36. 

\Factor : 

1.  a^—lab^^—x^.  4.  a'  +  6a5+9^>'— 4c^ 

2.  \-\-'lx-Vx^-if.  ^b.  l-x'-Sxy-Uf. 
^,  x^—1xy\y''—\,  6.  a;'+4a£c-y'  +  4a^ 


I^  ACTORS  115 

\y,  a'-l  +  10ab  +  2^b\  10.  l  +  Qax-a'-9x\ 

8.   -4-2ab  +  a'  +  b\  "^  9x'-ia'-9c'  +  12ac. 

K^  a'-^c'-Sab+lQb'.  12.  a'  +  2ab  +  b^  +  2cd-c'-d\ 

14.  a'  +  12m7i—10ab-4m'  +  2bb'-9n^. 

15.  ic2-12aa!-4^•y  +  36a'-4?/2-6^ 
X   XFactor,  and  simplify  the  factors : 

Al6.  (x  +  2i/y-(2x-^yy  19.-  (5a-3^)^  +  12«5-^>2-36a2. 

17.  (2x-Si/y-(^j-xy.  \20.  a^^-8a;?/  +  16/-(£c  +  4y)^ 

3^8.  (.^-5y)^  +  24£cy-9£c--16/.ai.  W0a'  +  20ab  +  b'-(a-2by, 

87.  Special  trinomials  of  the  type  form  x*+axy^+y. 

It  often  happens  that  trinomials  ol  the  form  x*-\  ax'^y'^+y* 
may  be  written  in  the  form  a^—¥  by  the  addition  and  subtrac- 
tion of  a  term.  The  addition  and  subtraction  will  not,  of  course, 
change  the  value  of  the  given  expression. 

Example.  1.     Factor  aj^  +  o^y  +  ^/^- 
Adding  and  subtracting  ar^i/^,  we  have 

x^  +  x^y^  +  y*=x^  +  2x^y^  +  y*—x^y^ 

=  (x''  +  yy-x'y' 

=  {x''  +  y^  +  xy)  {x''  +  y^—xy). 

Example  2 .     Factor  a?*  +  2x'^y''  +  9y\ 

Adding  and  subtracting  4x^y^,  we  have 

x*  +  2xY  +  ^y*=x'  +  Qx'y'  +  ^y*-4xY 

=  {x'  +  3yy-4xY 

=  (s(f  +  3y'  +  2xy){o(f  +  Sy''-2xy), 

Examples.     Factor  a?*— a?  V  + 2/*- 
Adding  and  subtracting  Sx^y^, 

x*-x^y^  +  t/=x*  +  2xY  +  y'-Sx^y' 

={x'  +  yy-BxY 

={a^  +  y'+xyy'S){oc'  +  y'-xyi/S). 


116  ALGEBRA 

Note. — These  factors  are  rational  in  x  and  y,  but  not  in  the  coeffi- 
cients. For  this  reason  the  expression  is  not  considered  factorable  in 
the  sense  in  which  the  term  is  usually  taken. 

Example  4.     Factor  4ic^—16a?*^*  +  92/^ 
Adding  and  subtracting  4:X*y*, 

4a?«— 16ic*2/*  +  ^y^=^o(f-12xY  +  9i/«-4^Y 
=  (2x*-3yy-4xY 

= {2x'  -  By'  +  2xY)  {2x*  -  3y'  -  2x'y') . 

EXERCISE  37.  4  ^ 

Factor : 

1.  x'-^-x'-i-l.  ^9.  36a*-76ay  +  25i/. 

^^.  l  +  Sx'  +  4x*.  10.  49a'-^19a'b'-\^4b\ 

3.  x'  +  x*  +  l.  J^^-  4iz;'-16i«y  +  25yi 

/  ^  4.  a*  +  2a'b'  +  9b*.  12.  04.Ty  +  12a;y+l. 

5.  a'-Sa'  +  9.  ^  }^-  «'-22aVy +  9a.'y- 

\«.  ,4£c*  +  16a!y +  25y*.  ^4.  4m*n«-45a^^^mV  +  25a*i«. 

"  lyi^  4a'  +  4a'b'  +  2^b\  />\^  16.  x''-^x'  +  U. 

V  8.  a;*-15a!y  +  9/.  vV  16.  25a*  +  16a'^»V  +  4^»V 

88.  Binomials  of  the  type  form  flf'4  6",  when /7  is  odd. 

By  §  77,  a"  +  b"  is  divisible  by  a^  b  mhen  n  is  odd.  Hence, 
the  factors  of  a"  +  b'\  when  n  is  odd^  are  a  +  b  and  the  quotient 
obtained  by  dividing  a"  +  6"  by  «  I  ^. 

By  division, 

a'+b'={a^-b){a'-ab  -\-b')', 
a'-\-b'=(a-^b){a'-d'b  +  a'b'-ab^  +6*) ; 
a''\rb''=:{a^b){a'-a'b  +  a'b'-a'b'-\^a'b'-ab'  +  b% 

Example  1.    Factor  x?  +  Sy^. 

X?  +  ^y^  may  be  written  a?^  +  {2yf^  and  therefore  may  be  divided 
by  x-[-2y. 


FACTORS 


11' 


Hence,  o(?-^Sy^={x-\-2y){3(?—2xy  +  4ty^), 
Example  2.     Factor  2>2w'  +  2436^ 
32a5  +  24365=(2a)H  (36)5 

= (2a  +  36)  j  (2a)*  -  {2a)\Zb)  +  (2a)'^(36)^  -  (2a)  (36)» + (36)* 
=  (2a  +  36)(16a*-24a^6  +  36a'^62-54a6H816*). 
Example  3.    Factor  a^ + 6^ 
a»  +  6»=(a»)3  +  (63)» 

=  (a«  +  6^)  {{ay-{a^){h^)  +  (6^)^} 
=(a3  +  6^)(a«-a='6H6«). 
But  aH6=^=(a  +  6)(a2-a6  +  62). 

Hence,       a^  +  lf={a  +  h){o}-ah  +  h''){a^-aW  +  W). 


<,  Factor : 
1.  x^-^y\ 
2.  Q^-\-yK 
/y^Z,  x'  +  y\ 
4.  a?+y\ 
^b.  x''  +  y'\ 
6.  27x^+a\ 
/X7.  Ux'  +  27y'. 

8.  125a^  +  216^>=^ 

9.  40a'  +  Ubb\ 
10.  432 +  2a;^ 


EXERCISE  38. 

0^11.  81a;^  +  3. 

^2.  ^'  +  1. 

13.  l  +  x\ 
h4:.  S2  +  x\ 
15.  l  +  3125a^ 

\6.  «W  +  1. 
,'\17.  «V  +  32. 


18. 


i25a;y. 


19.  J  +  1. 


1        7,5 


/^20.  ?"  -  1 

2/'       • 

21.  ^i^^'  +  ^V. 
^^2.  1024a'' 
23.-  12Sx^+^i^y\ 
24.  i^  +  2/. 

26.  ic^  +  l. 
58.  l  +  a'^». 


89.  Binomials  of  the  type  form  a'*— 6",  when  n  is  odd. 

By  §  77,  fl"  —  6"  *s  divisible  by  a — b  when  n  is  odd.  Hence, 
the  factors  of  w—b'\  when  n  is  odd^  are  a—b  and  the  quotient 
obtained  by  dividing  a"—b"  by  a—b. 

By  actual  division, 


X18  ALGEBRA 

a^-b^={a-b){a'-\-ab^-b') ; 
a'-b'  =  {a-b){a'^-a'b^-a'b'^-ab'^-b')\ 
a''-b'=^{a-b){a'  +  a'b^a*b'  +  d'b'+a'b'^ab'  +  b'). 
Example  1.     Factor  27x^—8?/^ 

27jL^—^y^  may  be  written  {^xf—i^yf,  and  consequently  may  be 
divided  by  Zx—2y.     Hence, 

27x'-8y'={Sx-2y)  {{'^x)^  +  {3x){2y)  +  {2yY} 
=  (Sx-2y){9x'  +  Qxy  +  ^y'). 

Example  2.     Factor  x^^—y^. 
x^^—y^={x^)^—y^ 

={x'-y){ix'y  +  {x'fy  +  {xYy'  +  {oc')y'  +  y*} 
=  {x^ — y)  {x^ + x^y  +  x*y'^  +  x^y^ + y*) . 


EXERCISE  39. 

Factor : 

1.  x^—y^.  11.  1— i^c^  \    gi    ^^     V^ 

>"2.  a^-h\  \'\2.  x'-\.                         '  ^~^' 

V3.  x'-y\  \  13.  32-y\  \  22.  «;^_'i'. 

^  5.  .-2/^  15.  .-32.^              )^^^  -^,_ 

N^  7.  8a^-2/^  Vl7.  l-a\  ,          ^^^, 

'^    8.  27a;^-64al  18.  128-a3y.  ^^6.  1^ — i. 


^ 


9.  125-34aal        >   19.  8l£c^-3.  i 

27    </' iL 

10.  8a«-^»\  20.  648-3^1  '  «'* 

28.  x^-\y\  29.  (a-*)•'^-(c-<7)^ 

30.  (a;-2y)5-(2a  +  ^>)l 

90.  Special  Binomials  of  the  type  form  a" +  6",  when  n  is  even. 
When  n  is  divisible  hy  Jf.^  hinomials  of  the  form,  a"  +  b''  may 
be  factored  as  in  §  87. 


FACTORS  119 

Example  1 .    Factor  x^ + a". 
Adding  and  subtracting  2a*x\ 

ix^  +  a^=3(^  +  2a* X*  +  a^— 2a*x* 
={x*  +  a*y-2a*x* 

=  {x*  +  a*  +  a^a?^/ 2)(.r*  +  a*-a''x'\/2). 
Note. — These  factors  are  rational  with  respect  ioX\\e  general  numbers 
X  and  a  but  not  with  respect  to  their  coefficients.     Factors  of  this  kind 
are  sometimes  useful. 
Example  2.     Factor  81a'Hl. 

=  (9a«)2  +  2(9a«)  +  l-2(9a«) 

=  {9a^  +  iy-2{9a^) 

=  (9a«  +  l  +  3a"^|/2)(9aHl-3a=^y'2). 

When  the  exponent  n  has  an  odd  factor^  binomials  of  the  form 
a"  +  b''mai/  be  loritten  as  the  sum  of  two  odd  powers^  and  fac- 
tored as  in  §  88. 

Example  1.     Factor  if-\-lf. 

={y'^h')  {{yy-{y')  {h')  +  m'} 
:={if+h')  {y'-h'y'+h*). 
Example  2.    Factor  x^"  + 1024. 
£c"  +  1024=(ir=')^  +  45 

={x^+4){(xy-{xy-4.+{xy-^^-{x:'ye+4*} 

=  (a?2  +  4)(.T«-4a?«4-16iC*-64a72  +  256). 


\           EXERCISE 

40. 

Factor : 
*^1.  a*-^b\ 

\ 

■'x  7.  x'*-\-l.      \ 

l\12. 

l  +  81a;^ 

Y2.  a'  +  b\ 

8.  x*  +  l. 

13. 

64a^«  +  729/. 

A    3.  a'-^b\ 
4.  a'^'^b'^ 

9.  03^°+ 1. 

14. 

X*     y* 
y'    ^'' 

^   5.  a^^-\-b'\ 

'Via  l+ar^ 

,„      1 

6.  a«+l. 

11.  16a;«  +  l. 

15. 

-'"+5r»- 

120  ALGEBRA 

91.  Binomials  of  the  form  a"—b'\  when  n  is  even. 

Binomials  of  the  form  w—b''^  when  n  is  even,  should  always 
be  written  in  the  form  a}—h\  and  factored  as  in  §  85. 

Note. — Factors  should  themselves  be  factored  when  possible. 

Example  1 .     Factor  a?* — ?/*• 

Writing  in  the  form  a^—lf^  we  have 

Factoring  x^—if,  =('«^  +  2/^)(<^  +  2/)('^— 2/)' 

Example  2.     Factor  729a«-64. 
Writing  in  the  form  a^—b^,  we  have 
729a*^-64=(27a=')^-8"'' 

=  (27aH8)(27a=^-8).  '. 

But  27aH8=(3a)H2^ 

=  (3a  +  2){(3a)2-(3a)-2  +  2=} 

=(3a  +  2)(9a2-6a  +  4). 
And  27a3-8=(3a)^-2=' 

=  (3a -2)  {(3a)- -f  (3a) -2  +  2--^} 

=  (3a-2)(9a2  +  6a  +  4). 
Hence         729a«-64=(3a  +  2)(3a-2)(9a2-6a4-4)(9a2  +  6a  +  4). 


EXERCISE  41. 

V) 

Factor : 

1.  a*-b\ 

8.  a'-b\ 

15.  l-x'\ 

2.  a'-b\ 

9.  a''-b\ 

16.  4-a;*. 

3.  a'-b\ 

10.  a''-b\ 

17.  16-a^^ 

4.  a''-b'\ 

11.  x'-l. 

18.  81-a;«. 

5.  a''-b'\ 

12.  x'-l. 

19.  100-a^«. 

6.  a''-b'\ 

13.  l-x\ 

20.  256a«-l. 

7.  a'-b\ 

14.  l-a;^". 

21.  81a'^-16a3'^ 

FACTORS  l^l 

22.  l-256a^^  26.  1024-a'».  29.  x^'-a"'. 


X 


/2«+4 


23.  6V-64a;«.  27.  -,-1.  ^0.  x'^'-^-if 

24.  xY-a'b\  ^     ^^4^,  31.  (a  +  by-c\ 
26.  16-81«^^>*.             ^^-  ^'~T6*                32.  (x-yy-(a-b)\ 

33.  a^  +  4a=^i  +  6a-^»^  +  4«^»^  +  ^»*-c\ 

34.  {2x-i/y-{a  +  Uy. 

92.  Polynomials  made  to  show  a  common  factor  by  grouping 
terms. 

The  terms  of  a  polynomial  may  often  be  so  grouped  as  to 
show  a  common  factor^  then  factored  as  in  §  81. 

Example  1 .     Factor  ax +ay  +  hx  +  by. 

Grouping  those  terms  which  show  a  common  monomial  factor, 
we  have 

ax+ay  +  bx+by=a{x  +  y)  +  h(x  +  y).    . 

This  is  a  binoinial  whose  terms  are  a{x-\-y}^nd  b{x+y),  and 
shows  a  common  factor  x  +  y. 
Hence  a{xA-y)  +  b{x+y)  =  {x+y){a  +  b). 

Therefore  ax  +  ay  +  bx+by=(x-\-  y)  {a-{-b). 

Example  2 .     Factor  x^  +  x'^—4x—4. 

x^+x'—4:X—x^x'^(x  +  l)—4ix  +  l) 
=  {x  +  l){x'-4:) 
={x  +  l){x  +  2)(:x-2). 
Example  3.    Factor  a^—¥—(a— bf. 

a-^-b'-{a-by=(a''-¥)-ia-bf 

=  {a-b){{a  +  b)-{a-b)} 
=  ia—b){a  +  b—a  +  b) 
=2b{a-b). 
Example  4.     Factor  a^—b^  +  ax— bx. 

a^—b^-\-ax—bx={a'—b^)  +  (ax—bx) 

=(a  +  6)(a-^&)  +  x(a— 5) 
={a—b){a  +  b  +  x). 


ALGEBRA 
EXERCISE  42. 


\ 


% 


^' 


/  l^ 


Factor : 

1.  ac-\-bc—ad—bd. 

2.  ax—bx—ay  +  bj/. 

X3.  ax-^bx  +  2a-^2b. 

4.  ax-\-4:X—ay—4:y. 
>\5.  a^b-^b-Va'c-^^c. 

6.  ax^'^-ax-\-bx^-^bx- 


k 


14.  £c— «+(»;— a)'. 

15.  a^x-\-a^y-^x  +  y. 
vVl6.  x^a-Vx'b-a-b. 

17.  j?/  +  3ic*-a;-3. 

18.  x^—x—y'^^y. 

19.  ax^—a—bx^  +  b. 


7.  2a-2^•  +  aJ-^>6^+2c  +  c(7.^^  20.  aW-a*-**  +  l. 


8.  ax—bx—a  +  b. 
Jl\9.  xy—x—y^\. 
10.  a^>-3a-25  +  6. 

^11.  a;^  +  aa;  +  ^i«  +  «5. 

12.  a;^  +  aa;-2a;-2a. 

13.  a;^— a^  +  C£c— «c. 

27.  aV 


>^^1.  ax^  +  bx—ax—b. 

K23.  a'-b'-{a'-by. 

24.  mnpq  +  2+pq-^2mn. 
'^'    25.  4:ax—2ay—2bx  +  by. 

26.  a^^>^-6^c^^-aW^  +  cW^ 
1-a^-a;^ 


— V^8.  aaj,?f  3air+5^2  +  15a;  +  2a  +  10. 
29.  a;^4-4a!^  +  4ic— aj^y— 4ajy— 4?/. 
ioO.  TTi^n*^ — my — q'^n^  +  (^y . 

31.  a'x'-ay-b'x'-i-by,  36.  a(a  +  c)-^>(^>  +  c). 

32.  x'a*-y'a*-b*x^  +  by.  37.  (a  +  ^>)2  +  5c(a  +  ^>) +6c'. 

33.  «V-a2  +  a^>i«=^-«7>.  38.  b'-c'  +  a(a-2b). 


34.  a;^ 


■2/^ 


■2^?^ 


2(xz-yw).    39.  aj*+£c^  +  y'-y*. 


35.  4a;y-(a;'  +  2/'-2')'.  40.  x'  +  x'z  +  xyz  +  y'z-y\ 

41.  2(l-a!)(l+a!)^+(l  +  a3'). 

42.  a^x  +  abx  +  ac  +  b'^y  +  aby  +  bc. 

93.  The  methods  of  the  foregoing  sections  will  enable  the 
student  to  factor  almost  any  expression.  Another  good 
method,  however,  based  upon  what  is  known  as  the  remainder 


FACTORS 


123 


theorem  is  discussed  in  the  following  sections.  This  method  is 
very  convenient  when  the  given  expression  cannot  readily  be 
thrown  into  one  of  the  familiar  type  forms. 

94.  The  remainder  theorem.  The  remainder  obtained  by 
dividing  a  polynomial  in  x  by  x  —  a  is  the  expression  that  would 
be  obtained  by  replacing  x  by  a  in  the  dividend. 

Thus,  dividing  x^-^-Zx—^  by  x—a, 


ic^'  +  Sr-S 
x^—ax 


x—a        (divisor) 


a?  +  (3-|-a)  (quotient) 


(3  +  «)a?— 5 
(3-|ra),r— 3a— g-^ 

a'^'  +  Sa- 5,  the  remainder,  which  is  the  same 
as  the  dividend  when  x  is  replaced  by  a. 

Dividing  i)(f—x^  +  dx+2  by  x—2^  we  have 


x^-  x^  +  3x+  2 

x-2 

x^-2x' 

x'  +  x  +  5 

x^  +  Bx+  2 

x'-2x 

5x+  2 
5£C-10 

12,  the  remainder 
Now  if,  in  the  dividend,  x  is  replaced  by  2,  the  dividend  be- 
comes i^,  which  is  the  reinainder. 

To  show  that  this  principle  is  true  in  general,  let  any  divi- 
dend be  represented  by  the  general  expression, 

Ax-  +  Bx'^-'~\'Cx^^-^+ -VMx  +  JSr. 

Let  the  quotient  and  remainder,  when  this  is  divided  by 
.c— a,  be  represented  by  Q  and  B^  respectively,  k 

Then,  by  §  51,      <l%  KjJ>M^  «  \m3<M^ 
Ax--\'Bxn-^'{-Cx--K -^Mx^-]Sr={x-a)Q-^B. 

Tn  this  identity  a?  is  a  general  number.,  and  may  have  any 
value,  hence  the  identity  must  be  true  when  x=a. 


124    "  ALGEBRA 

But  when  x—a^  it  becomes 

Aa^  '^Ba'^-^^-Car-'^- -\- Ma-\-N'={a-a)Q-\- B, 

for  B  does  not  contain  a  term  in  x  and  is  not  affected  by  the 
change. 

Since  {a—a)Q  is  zero,  therefore 

which  establishes  the  general  principle. 

Example  1.  Find  the  remainder  when  x*—2x^-\-Z  is  divided  by 
x-2. 

'     Replacing  x  by  2,  x'-2x''  +  ^  becomes  2*-2-2'  +  3=ll. 
Hence,  the  remainder  is  11. 

Example  2.  Find  the  remainder  when  x?—4.x'^—2x  +  2Q  is 
divided  by -^  +  3. 

Here  a^  +  3,  written  in  the  form  a?— a,  is  a?— (—3). 
Replacing  ic  by  —3,  x^—4:X^-~2x+20  becomes 

(-3)=^-4(-3)2-2(-3)  +  20=-37,  the  remainder. 

95.  Factoring  by  means  of  the  remainder  theorem. 

Apolynoniial  is  exactly  divisible  by  x — a  lohen  the  remainder 
is  zero.  From  §  94  it  follows  that  Ax''  +  Bx''-'+  Cx''-'+  •  •  •  • 
-\-Mx+JV  is  divisible  by  x—a,  if  Aa''-hBa''-^+  Ca''-^+  •  •  •  • 
-hMa+JV=0. 

That    is,    Ax^'-^Bx^'-'+Cx''-''^   •  • -}-Mx  +  ]^  will 

contain  the  factor  x—a,  if,  when  x  is  replaced  by  a,  the  polyno- 
mial becomes  0. 

This  principle  will  enable  us  to  find  the  binomial  factors  of 
many  polynomials  that  do  not  come  under  the  simple  type 
forms. 

Example  1 .    Factor  xH  4^ — 5 . 

We    look    for    a    number  which    will  make  the  expression 


FACTORS  125 

x^  +  4x—5  equal  to  zero,  when  x  is  replaced  by  the  number.  Since 
the  product  of  the  second  terms  of  the  two  factors  is  —5,  we 
should  try  -j-1  and  -j-5. 

It  is  easily  seen  that  1  is  the  number;  for  whenic=l,  x^  +  4x—5= 
1^  +  4-1  —  5=0.     Hence  x—1  is  a  factor. 

Dividing  x'^  +  4:X—5  by  ic— 1,  x-{-5  is  found  to  be  the  other  fac- 
tor. This  factor  might  also  have  been  discovered  by  use  of  the 
remainder  theorem. 

Example  2.     Factor  x^  +  ix—4:X—16. 

It  is  seen,  by  trying  numbers,  J^l,  :^2,  -|-4,  etc.,  that  when  x 
is  2,  —2,  or  —4,  this  expression  becomes  zero.  Hence,  a?— 2, 
a? +2,  and  x  +  4  are  factors.  These  are  all  of  the  factors,  for  the 
product  must  give  a  term  or*  which  is  the  highest  power  in  the 
expression. 

Then,  ^  +  4.z^^-4.r-16=(a!-2)Gx+2)(x  +  4). 

Note  that  this  expression  could  have  been  factored  easily  by 
the  method  of  grouping  terms. 

Examples.     Factor  ar'—Saj^  +  lOeT— 12. 

This  becomes  zero  when  x= 3.  Hence,  a?— 3  is  a  binomial  factor. 
Dividing  the  expression  by  x— 3,  we  get  for  quotient  x^—2x-\-4:,  a 
trinomial  factor.  If  the  factor  x'^—2x+4  had  rational  factors, 
they  would  be  obtained  by  the  method  of  §  83.  It  has  no  rational 
factor. 

Hence,  x^-^x''  +  10x-12={x-2>){x^-2x+4). 

Note. — In  attempting  to  find  a  value  for  x  that  will  reduce  the  given 
expression  to  zero,  it  is  wise  to  begin  by  trying  1  or  — 1  ;  then  try  larger 
possible  numbers  until  the  required  number  is  found. 

Example  4.     Factor  o(?-\-2x^y—xy'^—2y^. 

This  expression  becomes  zero  when  x=y.  Hence,  x—y  is  a 
factor.     Similarly  a? +  2/ and  a? +  22/ are  factors. 

Hence,        a^  +  2x''y-xy^-2y^={x-y)(x  +  y){x+2y). 


126  ALGEBRA 

Example  5.  Show  that  a +  6  is  a  factor  of  a" +  6"  only  when  n 
is  odd,  and  thus  prove  {d)  of  §  77. 

Putting  —b  for  a  in  a" +  6",  we  get  (—6)" +  6".  If  n  is  odd, 
(— 6)"  +  6"=— 6"  +  6"=0.     If  w  is  even,  (— 6)"  +  6''=6'^H-6''=:26». 

Hence,  a  +  &  is  a  factor  of  a"  +  6"  when  n  is  odd  and  not  when  m 
is  even. 

When  a  factor  of  the  form  x—a  has  been  found,  the  work 
of  dividing  it  out  may  be  abridged.  The  method  may  be 
derived  from  the  following  example  : 

Divide  ar*— 5a;^  +  5x— 7  by  £C— 3. 


ar»— Sar'  +  Sic— 

7 

x-S 

a^-3x^ 

a;'— 2a;— 1 

-2o(f 

-2x'  +  6x 

—  X 

—x-\- 

3 

10,  the  remainder. 

Observe  :  (l)  that  the  coefficient  of  the  first  term  of  the  quotient  is 
the  coefficient  of  the  first  term  of  the  dividend,  and  the  succeeding 
coefficients  are  the  coefficients  of  the  first  terms  of  the  succeeding 
remainders. 

{2)  That  since  the  first  term  of  each  partial  product  caficels  the 
term  of  the  dividend  immediately  above  it,  the  first  term  of  each 
partial  product  need  not  have  been  written. 

(3)  That  if  the  sign  of  each  term  of  the  partial  products  had 
been  changed,  this  term  micfht  have  been  added  to  the  corre- 
sponding term  of  the  divide^id.  (This  can  be  done  by  changing 
the  sign  of  each  term  of  the  divisor.) 

(^)  That  if  the  dividend  is  written  in  descending  powers  of  the 
first  term  of  the  divisor,  this  term  of  the  divisor  need  not  be  written. 

Omitting  all  work  that  is  unnecessary,  writing  the  coefficients 


FACTORS  X27 

only,  and  raising  each  number  of  the  several  remainders  into  the 
same  line,  the  work  may  be  written  as  follows  : 

1-5  +  5-7  |3_ 

+3-6-3 
1-2-1,-10  where  1,  —2,  and  —1  are 

the  coefficients  of  the  quotient,  and  — 10  is  the  remainder. 
This  abreviated  form  of  division  is  called  synthetic  division. 
EX4.MPLE  1.     Divide  2x^—4x^  +  7x+23hy  x  +  S. 

2-  4+   7  +   23  1-3 

-  6  +  30  -111 
2-10  +  37,-  88 

Hence  the  quotient  is  2xf—10x+S7  and  the  remainder  is  —88. 

Observe  that  the  first  coefficient  of  the  dividend  is  brought  doivn 
for  the  first  coefficient  of  the  quotient. 

This  number  is  multiplied  by  the  number  in  the  divisor  and 
added  to  the  next  term  of  the  dividend  for  the  second  coefficient 
of  the  quotient,  and  so  on  to  the  last  term. 

Example  2.     Divide  2a?*— 10x^-13  by  x—2. 

2  +  0-10  +  0-13  |2 
+  4+   8-4-  8 
"^  2  +  4-2-4,  -21 

Hence,  the  quotient  is  2xr^-\-4iXi^—2x—4:  with  a  remainder  of— 21. 
Observe  that  where  any  power  of  x  is  ivanting  its  place  must 
be  supplied  by  a  zero. 

Formulate  a  rule  for  the  process. 

Perform  the  following  divisions  by  the  synthetic  process : 

1.  x^  +  Zx'-lx-Z  by  a;-2.  6.  x'-la?-\-2x-12  by  x-\-2. 

2.  3a;^  +  7a;^-3a;  +  15  by  aj+3.        7.  x'  +  Zx'  +  x—l  by  ic-3. 

3.  x'-^x'^lx-l  by,£c-2.  8.  2a;''+jc=^-7  by  a;  +  3. 

4.  2iK*— 9a;'  +  3a;— 5  by  cc— 4.  9.  x'-i-hx'^-x—^  by  a;  +  4. 

5.  3a;*-7a;^  +  13  by  cc  +  2.  10.  a;*  +  3aj^-9  by  x-2. 

The  pupil  should  now  be  able  to  factor  any  factorable  ex- 
pression that  is  likely  to  occur  in  elementary  algebra. 


128  ALGEBRA 

The  following  general  suggestions  may  be  useful : 

1.  First  remove  all  monomial  factors. 

2.  Try  to  bring  the  resulting  polynomial  under  some  one  of 
the  binomial  or  trinomial  types. 

3.  If  imsuccessful^  try  the  remainder  theorem. 

Jf..  Be  sure  that  all  factors  are  prime.  ^         s 

\^' 
EXEBCISE  43. 

vi  By  the  use  of  the  remainder  theorem,  factor  the  following : 


1. 

Ix^-x-l. 

6. 

'M- 

-9.^  +  9. 

9.  x'-^x'-Sx-2. 

2. 

3^^  +  5cc  +  2. 

6. 

3a3^  +  14a!+15. 

10.  x'-{-7x'i-7x-16. 

3. 

2£c2-5ic  +  2. 

7. 

Zx'- 

-13^-10. 

11.  x*  +  x'-2x-2. 

4. 

^x^-^lx-1. 

8. 

x'- 

-Qx'  +  Ux- 

-6.  12.  x^  +  Sx'-i. 

13. 

2a;-"'  +  3i«'-50i«  +  24. 

19.  a'- 

-a'  +  a'-l. 

14. 

^x'-1x^'\-x- 

-2. 

20.  2x' 

-Sxy  +  y\ 

Jl5. 

a'  +  3a'  +  3a- 

+  2. 

21.  ^x' 

+  2xy-y\ 

16. 

rt^  +  2a-'-4a- 

-3. 

22.  x'-- 

h  2x'^i/ —xii^ — 2?/. 

17.  a^-21a-20.  23.  x^-2x^y-^xy''-^\%y\ 

18.  a*  +  7a=»  +  13a^  +  7a  +  12.  24.  a^^^a^b^ab'-W. 

25.  Show  that  w—b"  is  akoays  divisible  by  a—b,  and  thus 
prove  (a)  of  §  77. 

26.  Show  that  «"—?>>"  is  divisible  by  «  +  ^  only  when  oi  is 
even^  and  thus  prove  {b)  of  §  77- 

27.  Show  that  a"  +  ^"  is  never  divisible  by  a—b,  and  thus 
prove  (c)  of  §  77. 

By  any  method  of  this  chapter,  factor  the  following : 

28.  ax'-Q>^a\  31.  4a^  +  32«^.  +  39Z>l 

29.  x'+--2.  32.  {x-\-iy-bx-2^. 

30.  ^ahx^  +  2abx-ab.  33.  ax'-Wx^-haK 


FACTORS  129 

34   ^  +  ??  +  l  ^^*  4«*  +  3a;y  +  9y*. 

'  a'      a       '  ,60.  a'b-bc'  +  a'c-c\ 

35.  xy^-z'-xz-yz.  '  ^^    a.3_^2_4^_g^ 

36.  x'+^x'^-Qx+n.  62.  a'-Va?-la-^. 

37.  4(a-6y-(y  +  2)l  63.  x^+x'-VJx^U, 

38.  9a;^-27a;y  +  20y^  64.  a;^-ll£c^  +  31a;-21. 

39.  a'  +  aW-{-b\  65.  £c=^-6£c''  +  lla;-6. 

40.  cc*— ISicy  +  y*.  66.  a;='  +  2a3'-9a)-18. 

41.  a;6  +  a;='-42.  67.  2/='-92^  +  25a;'^-10a;y. 

42.  ^*+2xy-35/.  68.  Qx^'^  +  x^r-lbf^ 

1     9  69.  3(a-^)'-14(a-^>)  +  8. 

^^*  ^'"^a*"!*  70.  7(a  +  ^)'-llc(a  +  ^>)-6c2. 

44.  x'^-^x'-x'-^x.  71.  £c*-aj^^-17x^  +  5a;  +  60. 

45.  l  +  ^»y-(a;'  +  Hi/^  72.  a^-a'b-a^h. 

46.  (aj^'+y'-s'O'-^a^y-  73.  x'-%x-ax-V^a. 

47.  x'-2xy'-y'-\-^x^y.  74.  aa;^-3aV  +  2a'a;. 

48.  a;/  +  7a;y-30a;.  sy75.  2£c^-3x^-9a;  +  14. 

49.  ac-\-cd—ab—bd.  76.  35a;'— 74a; +  35. 

50.  iK*-(a;-6)l  77.  a;^  +  a;*-56a-^ 

51.  72«'  +  41^-45.  78.  x'-^bx-^^-W. 

52.  a'b'-a'-b'-\-l.  79.  a;^  +  7x^-5a;'-35.  t^ 

53.  2s'  +  5s«!-12^^  80.  a;^  +  4£c'4-a;-6. 

54.  6a='a;-aV-9a*.  81.  £c=^-3a;'  +  7£c-21. 

55.  bcx-\-acx^-\-bx^-\-ax'^.  82.   a;3  4-25a;2_[-8a;-16. 

56.  a*-2(^»'  +  c')a'  +  (52  +  c')l  83.  10  {x-\-yy  +  lz{x^y)-^z\ 

57.  a;'-(4a='  +  ^>')a'£c  +  4a''6l  84.  24(a+&)2-5c(a  + 6)— SGc^. 

58.  a;'  +  (a-%-2a(a+5).  85.  a;'+4ic+4-4a'^+4ay-yl 

9 


CHAPTER  X. 
COMMON  FACTORS  AND  MULTIPLES. 

96.  Integral  and  fractional  terms.  A  fractional  term  is  a  term 
which  contains  one  or  more  general  numbers  in  the  divisor. 
Otherwise,  the  term  is  integral. 

Thus,  r,    ^^7      2a  ^   ^^^  ct-^{^—y),  ^Y^f^CLctional  terms,  for 

there  are  general  numbers  in  the  divisor.     And  a^,    :^x^y^   ^^- , 

and  —  |a&,  are  integral  terms,  for  the  divisors  are  not  general 
numbers.  A  fractional  term  may  become  a  whole  number,  and 
an  integral  term  a  numerical  fraction,  when  definite  values  are 
assigned  to  the  general  numbers  involved.  Hence,  the  classifica- 
tion of  terms  into  integral  and  fractional  terms  has  no  reference 
to  arithmetical  whole  numbers  and  fractions. 

97.  An  integral  expression  is  an  expression  whose  terms  are 
all  integral. 

Thus,  ^a^—§ab  +  b^  and  Soc^  +  2x'^  +  x  + 1  are  both  integral  ex- 
pressions. 

Observe  that  the  ^  and  |  do  not  make  the  terms  fractional. 
The  nature  of  a  term  depends  upon  its  general  numbers. 

A  fractional  expression  is  an  expression  whose  terms  are  all 

fractional. 

2«]t^     6?y^    z^         abc     x^ 
Thus,  — — -J-  +  —  and  n^-  4-  rr  are  fractional  expressions. 

An  expression  may  be  integral  with  respect  to  certain  general 
numbers  and  fractional  with  respect  to  others.     That  is,  certain 

130 


COMMON  FACTORS  AND  MULTIPLES  131 

general  numbers  appear  in  the  dividend  only,  while  others 
appear  in  the  divisor. 

Thus,  ^—2^  +  -^  is  integral  with  respect  to  a,  hwi  fractional 
with  respect  to  y. 

A  mixed  expression  is  an  expression  some  of  whose  terms  are 
integral  and  some  fractional. 

3c^  a^     a 

Thus,  —  +  a  and  2ab  +  ^  "^  3x  ^^^  wi«a?ed  expressions.  The  sec- 
ond, however,  is  integral  with  respect  to  a. 

98.  The  degree  of  a  rational  integral  term  is  the  number  of 
literal  factors  in  it.  This  amounts  to  the  sum  of  the  exponents 
of  all  letters  in  the  term. 

Thus,  the  degree  of  Qa^b  is  3.  The  degree  of  loc^y^z^  is  6.  And 
4abcde  is  of  the  fifth  degree.  The  term  5a^a^  is  of  the  third 
degree  in  x.     And  ab^x*y^  is  of  the  seventh  degree  in  x  and  y. 

The  degn^-ee  of  a  rational  integral  expression  is  the  same  as 
the  degree  of  its  ternj  of  higliest  degree. 

Thus,  5x^—x-{-4:  is  an  expression  of  the  third  degree.  And 
2x^y^—Sxy-{-l  is  of  the  fourth  degree.  This  expression  is  of  the 
second  degree  in  x.     The  expression  is  of  what  degree  in  y  ? 

An  expression  whose  terms  are  all  of  the  same  degree  is 
called  a  homogeneous  expression. 

Thus,  5x'^—2xy  +  y^  is  a  homogeneous  expression. 

99.  The  highest  common  factor  of  two  or  more  expressions 
is  the  expression  with  greatest  numerical  coefficient  and  of 
highest  degree  that  will  exactly  divide  each  of  them. 

Thus,  the  highest  common  factor  of  a^6V  and  aWc^  is  a^b^c^. 
And  the  highest  common  factor  of  lOor^T/z*  and  Sx^y^z^  is  2o(?yz^, 

It  follows  from  the  definition  above  that  the  numerical  co- 
efficient of  the  highest  common  factor  is  the  greatest  number 


132  ALGEBRA 

that  will  exactly  divide  the  numerical  coefficient  of  each  ex- 
pression ;  i.  e.,  the  greatest  common  divisor  of  the  numerical 
coefficients.  By  the  coefficient  of  a  polynomial  is  meant  its 
numerical  factor.  Thus,  2  is  the  numerical  coefficient  of 
2a;2-6i«  +  4  or  of  2(£c^-3a^  +  2). 

Also  the  highest  power  of  any  literal  factor  that  will  exactly 
divide  each  expression  is  the  lowest  power  of  that  factor  found 
in  any  one  of  the  expressions. 

A  factor  which  does  not  appear  in  every  one  of  the  expressions 
can  not  appear  in  the  highest  common  factor.    Hence  the  rule  :* 

To  form  the  highest  common  factor  of  ttoo  or  more  m^onomials^ 
take  the  product  of  the  greatest  common  divisor  of  their  numerical 
coefficients^  and  each  letter  raised  to  the  lowest  power  to  which  it 
appears  in  any  of  the  expressions. 

In  polynomials^  factor  each  expressio7i  qnd  proceed  as  with 
monomials^  treating  each  factor  as  yon  woidd  treat  a  single 
letter. 

Example  1.     Find  the  highest  common  factor  of  12a?^i/^  and 

The  greatest  number  that  will  exactly  divide  12  and  16  is  4, 
their  greatest  common  divisor.  The  highest  power  of  x  that  will 
divide  both  a^  and  x^  is  a?%  the  lowest  power  present.  And  the 
highest  power  of  y  that  will  divide  both  y^  and  y^  is  ?/^,  the  lowest 
power  present.  Since  z  does  not  appear  in  both  expressions,  it 
can  not  appear  in  the  highest  common  factor.  Hence,  the  highest 
common  factor  is  the  product  Ax^y^. 

*  There  is  a  method  of  obtaining  the  liighest  common  factor  of  two 
expressions  without  resolving  them  into  their  factors.  It  is  similar  to 
the  EucUdean,  or  long  division,  process  sometimes  employed  in  arith- 
metic of  finding  the  greatest  common  divisor  of  two  numbers.  The 
discussion  of  this  method  is  found  in  the  Appendix.  This  method 
is  seldom  used  in  practice.  The  factoring  process  here  discussed  is 
sufficient  for  our  purpose. 


COMMON  FACTORS  AND  MULTIPLES  133 

Example  2.  Find  the  highest  common  factor  of  SOa'b^c, 
12a^¥c\  and  18d'b*c\ 

The  greatest  number  that  will  divide  30,  12,  and  18,  is  6.  The 
lowest  powers  present  of  a,  6,  and  c,  are  a^,  6^  and  c,  respectively. 
Hence,  the  highest  common  factor  is  6a'*6^c. 

Example  3.  Find  the  highest  common  factor  of  iK^— 6^— 27 
andx'  +  6a?+9. 

x^-ex-27={x  +  3){x-9);  x''  +  6x  +  9={x-\-3Y.  The  lowest  power 
of  ic  +  3  present  is  the  first  power,  x  +  S.  And  x—9  is  not  a  com- 
mon factor.     Hence,  the  highest  common  factor  is  a? +  3. 

Example  4.     Find  the  highest  common  factor  of 
a^x—a^bx—Qab^x^  a^bx^—4ab^x'^  +  S¥x^,  and  a^x'^—2a^bx^—Sab'^x^, 

We  have  a^x—(i%x—Qa¥x=ax{a—3b){a  +  2b); 

a''bx'-4ab^x''  +  3b^x''=bx-{a-Sb)ia  -b); 
a^x^-2a%x^-Sab''x^=ax\a-3b){a  +  b). 

Hence,  highest  common  factor  =x{a  —  ^b) 

=ax—Sbx. 

In  finding  the  If.  C.F.  of  two  or  more  expressions  when  hut 
one  of  the  expressions  is  easily  factored  by  inspection,  we  may 
use  the  most  likely  factors  of  the  factored  expression  as  divisors 
of  the  other  expressions. 

Example  5.  Find  the  highest  common  factor  oix^—Zx-\r2  and 
ar*  — 4x^  +  4a?— 1. 

Qi?—Zx-^2—{x—\){x—2).     Now  since  —2  of  the  second  factor  is 
not  a  factor  of  —1  of  the  second  expression,  the  "most  likely 
factor"  is  x—\  which  by  trial  is  found  to  be  a  factor  of 
x'-^x'^^x-X.     Hence,  x-1  is  the  H.C.F. 

Example  6.  Find  the  highest  common  factor  of  2x!^+Sx—2 
Sind4x^-\-lQx'-19x  +  5. 

2x^  +  3x—2={x  +  2){2x—l).  Herethemost  likely  factor  is  2a?— 1. 
Why  ?  By  trial  2x—l  is  found  to  be  a  factor  of  the  second  ex- 
pression and  hence  is  the  H.C.F. 


134:  ALGEBRA 

EXERCISE  44 

Find  the  highest  common  factor  of : 
'   1.  ^a'b\  Mb\  13.  SGa^Sys,  72xyz,  lSOxy'z\ 

2.  9a*!}'c\  12a'b'c\  ^14.  17baWc\  70a'b%  10ba*bcP. 

3.  UxY,  SOxyz\  15.  24m'na\  42ma\  ISm'a'b. 

4.  Sa^yV,  9Qxy\  ^16.  15aV?i^  40«Vwi,  35aV//2^ 
/>t.  a'b\  10a'bc\  1 17.  ^ba'b\  bQa'b\  9SaW. 

6.  2a; V^',  y^^-  ^18.  -98^;^',  21a;y^,  -28a;yV. 

^7.  axy,  Sbxy.  -     19.  acc^y^,  bx*y\  cxy,  2ic^y^ 

j8.  49a^>Vy,  21a'¥cx\  ^20.  2(a  +  ^>)^  4(rt+^»)\ 

^9.  ^^xy,  -bQxi/z\  21.  14(a;-y)«,  21(i«-y)*. 

i>^0,  QSab'c^d\  -2>d'bd\  22.  16(a  +  a.')VS6(a  +  a3)'(^>  +  .y)\ 

11.  ^xYz,_^xYz\  2xy^z\  23.  (cc-1)^  {x-l){x-\-1). 

v^l2.  2Sa''b'c'd,  Qa'b\  lQa'b'c\         24.  10(a;  +  l)^  5(a!-l)2(ic  +  l). 
^5.  39(a;-l)-Xa3+l)^  26(a;+l)(a^-l)*. 
j   26.  bQ{x-l)(x  +  2)(x  +  S),  (x  +  2y{x-^). 
"?^27.  ic^-1,  a;^-l. 
■   28.  8iB='  +  l,  (2x  +  l)(x-S). 
29.  4(a;  +  l)',  20(a;-l)(a;+l),  36(ic-l)^ 
yio,  x'-l,x'-dx  +  2.  Z4:.  2x\x'-Sx,x'  +  4x. 

31.  x'-l,  x'  +  bx-Q.  \}^5.  x'-Sx,  9-x\  x'-^x  +  Q, 

32.  a'-b\  a'-b\  36.  a^^  +  l,  a;=^  +  l,  tc  +  l. 
^3.  a'-2ab  +  b\  (a-b)(a  +  2b).   37.  2a;'^  +  a;-6,  6if^-7ic-3. 

38.  3aa;' -  Sa\  4a V  +  2a V -  Qa'x.      • 

39.  a^ - ab\  a'  +  a^^*  +  a^>  +  b\ 

^^40.  a■^  +  3a2a;  +  2aa;^  a*  +  6a^ic  +  8a V. 

41.  a^-9a='  +  26a-24,  a''-12a2  +  27a-60. 
^  42.  7m^-2m2-5,  7m='  +  12mM  lOm  +  5. 


COMMON  FACTORS  AND  MULTIPLES  135 

43.  a'  +  ^ab  +  2b\a'  +  ^ab  +  4:b\  a}-^ab~U\ 
/  44.  1— £c^,  1— ic^,  x—x". 

46.  Vla?  —  l^ab^Zb\  ^w'-^a^b^-'lab^—W, 

46.  m'^  +  m— 6,  m='— 2m'— m  +  2,  m='  +  3m'— 6m— 8. 

47.  ic='-3a;y'-2y\  o?-x'y—^y\ 
JC^S.  a*  +  b\  a^-b\  a'  +  aby^+b\ 

49.  a=^-a'-5a— 3,  a^-4a'^-lla-6. 

60.  l-x%  l-2£c=*  +  a;«,  l  +  a;  +  £c^ 

61.  a'—bax  +  ^x^  d^—a^x  +  Zax'^—Sx^. 

62.  ic'-7a;  +  10,  4£c='-25a;'  +  20a;  +  25. 

63.  6x'(/  +  ^xf-2i/\  Sx'  +  AxSj-4x}/\ 


i 


100.  A  common  multiple  of  two  or  more  expressions  is  an 
expression  which  is  exactly  divisible  by  each  of  them. 

Two  or  more  expressions  may  have  an  indefinitely  great 
number  of  common  multiples. 

Thus,  a  few  common  multiples  of  2o(^  and  3xy^  are  607^1/^,  6x^1/^^ 
ex*y\  6x^y\  Qx'^if,  12x^y\  18x^y\  24x'''y\  etc.  Each  of  these 
expressions  is  divisible  by  both  2x^  and  3xy^. 

The  lowest  common  multiple  of  two  or  more  expressions  is 
the  expression  with  the  least  numerical  coefficient  and  of  lowest 
degree  that  can  be  exactly  divided  by  each  of  them. 

It  follows  that  the  numerical  coefficient  of  the  lowest  com- 
mon multiple  is  the  least  number  that  can  be  exactly  divided 
by  the  numerical  coefficient  of  each  expression ;  i.  e.,  the  least 
common  multiple  of  the  numerical  coefficients.  And  the  lowest 
power  of  any  literal  factor  that  can  be  divided  by  each  expres- 
sion is  the  highest  power  of  that  factor  found  in  any  one  of  the 
expressions.    A  factor  appearing  in  any  one  of  the  expressions 


136  ALGEBRA 

must  appear  in  the  lowest    common    multiple.    Hence   the 
rule :  * 

To  form  the  longest  common  multiple  oftico  or  more  monom,ials^ 
take  the  product  of  the  least  common  multiple  of  the  numerical 
coefficients^  and  each  letter  raised  to  the  highest  power  to  v^hich  it 
appears  in  any  one  of  the  expressions. 

In  polynomials^  factor  each  expression  and  proceed  as  vnth 
monomials^  treating  each  factor  as  you  icould  treat  a  single  letter. 

Note. — To  find  the  least  common  multiple  of  two  or  more  numbers, 
as  in  arithmetic,  separate  them  into  their  prime  factors,  and  take  each 
prime  factor  the  greatest  number  of  times  that  it  occurs  in  any  one 
of  the  numbers.  Thus  for  72  and  96,  we  have  72=2-2-2-3-3,  and 
96=2-2-2-2-2-3.  Hence,  the  least  common  multiple  of  72  and  96  is 
2-2-2-2-2-3-3^288. 

Example  1.  Find  the  lowest  common  multiple  of  2^aWc^  and 
SOa'b'c. 

The  least  number  divisible  by  24  and  30  is  120.  And  the  highest 
powers  of  a,  6,  and  c,  in  the  two  expressions,  are  (i\  b^,  and  c*, 
respectively.     Hence,  the  lowest  common  multiple  is  120a*6V. 

Example  2.  Find  the  lowest  common  multiple  of  15aV,  21y^z, 
3&a''z\ 

Here  the  least  common  multiple  of  15,  21,  and  36  is  1260.  The 
highest  powers  of  a,  y,  and  z  are  a\  2/^,  and  s^,  respectively. 
Hence,  the  least  common  multiple  is  1260a*2/"^^ 

^Example  8.     Find  the  lowest  common  multiple  of  2x'^—4x  +  2, 
\^'  +  x-2,  and  5x'-^10x. 

*  It  is  easily  shown  that  the  product  of  the  highest  common  factor 
and  the  lowest  common  multiple  of  two  expressions  equals  the  product 
of  the  expressions.  Hence  the.  lowest  common  multiple  of  two  expres- 
sions may  be  obtained  by  dividing  the  product  of  the  expressions  by 
their  highest  common  factor.  Since  this  method  could  be  of  value  only 
in  case  the  factors  of  the  expressions  could  not  be  found,  it  is  not  used 
in  this  book.  ' 


COMMON  FACTORS  AND  MULTIPLES  137 

Here  2x'-4x  +  2=2{x~iy; 

x'  +  x-2={x  +  2){x—l); 
5x''  +  10x=5x{x-\-2). 

The  least  common  multiple  of  2  and  5  is  10.  The  highest  power 
of  the  factor  xia  x;  of  x—1  is(a?— 1)^;  and  of  a? +  2  is  a? +  2.  Hence, 
the  lowest  common  multiple  is  10a7(a?— l)^(a?  +  2). 

/ 
EXERCISE  45. 

Find  the  lowest  common  multiple  of : 

1.  ^a'b,  Sab\  7.  2bm'n\  4:cmi7i\  10a'm7i\ 

2.  Qa'b\  10ab\  8.  15^//,  6pY,  Aj^Y- 
'     3.  M'bc\  ^ab\                           y  9.  SQa'%  9a'b%  lQd'b\ 

y4.  Sla!^',  l^x^f.  10.  27x''i/,  QxY,  4xi/\  ^xHj\ 

6.  lxy\  ^x%  GccV.  11.  {x-\-y)\  2{x  +  y)\ 

b^.  2la'bc\  4a'bc\  Qab\  12.  (a-b)\  S{a-b)(a  +  by. 

>  13.  x{x—l)(x  +  l),  2x'(a;  +  l). 
14.  x'(l-{-xy(l-x),  (l  +  a;)(l-a;). 
^\6.  10(a  +  ^»)'(c-J),  15(c-c7)^(a  +  J). 
^%6.  25rt^(a-^>)(2a  +  ^)^  45aa!(2a  +  ^)l 

17.  aj'^-l,  x'-2x  +  l.  :  23.  2a;^  +  3a;-2,  3£c^  +  7a!  +  2. 

18.  x'-l,  x'-x-2.  24.  2iK2-aj-10,  2x'  +  x-S. 
y  19.  l-a;^a;+a;^-2ar^                     2,5.  2a;=^  +  2a;,  aj=' +  5a;  +  6. 

]M0.  a'-b\  2a'  +  ^ab  +  b\  26.  a;^-l,  x'-x,  2x\ 

21.  a2  +  5a  +  6,  «^  +  7«  +  12.  27.  x'-l,  a;^  +  3a;  +  2,  a;^  +  a;-2 

c/22.  x'  +  x-SO,  a;'  +  5£c-6.  28.  l-x\  l-x\  1-x. 

29.  x'-fii^,  2x'-4x,  x'+  1. 
'  30.  a;^  +  5a;-14,  4a;^-16a;^  +  16a;,  ^x\ 
131.  a'-b\  a^-b\  a'-b\ 
^^32.  l-x\  l-x\  l-a;«. 


138  ALGEBRA 

33.  x  +  l,x^  +  l,x'-l. 

34.  x'  +  2x'-Sx,  2x'  +  Qx''  +  2x  +  6,  Sx^—^x\ 

35.  a*-b\  a*  +  2aW  +  b\  a'-2aW  +  b\ 

36.  27a=^-8,  9a'-4,  9a'-12a  +  4. 

37.  x-x\  lOx  +  dx'-x^  x-x'-{-x^-x*, 

38.  05^-1,  x'  +  x-2,  x'  +  bx  +  Q. 

39.  x'-l,  x'  +  x'  +  x,  Sx\ 

^40.  a'-l>'\  a'  +  a'b-ab'-b\  a'-a'b-ab'  +  b\ 
lAi,  ic2-15a;  +  36,  aj='-3aj2-2a3  +  6. 

42.  a^-a^  +  a  +  S,  a^  +  a'-Sa^-a  +  S. 

43.  6a^  +  a'^-5a-2,  6a^  +  5a^-3a-2. 

44.  ^x^  —  lx'y—2xij\  ^x^  +  xy—4y\ 

45.  a?*-13a;'  +  36,  aj*-a3='-7a;'  +  a;  +  6. 
^46.  633^-03-1,  2a;M-3a^-2. 

47.  4a;^-4£c4-l,  4a;2-l,  4a;2  +  4a;  +  l. 
I    48.  ax—ay  —  bx  +  by,  x'^—2x7/  +  y^. 
I     49.  6a;^  +  13aj-28,  12a^2-3l£c  +  20. 
I     60.  8a;^  +  30a;  +  7,  12a;^-29a;-8. 


CHAPTER  XI. 
FRACTIONS. 

101.  In  §  52,  §  53,  §  54,  §  55,  some  principles  of  fractions 
were  established  which  the  student  should  now  review.  Other 
principles  of  fractions  in  common  use  in  the  treatment  of 
algebraic  expressions  and  equations  are  here  given. 

102.  The  algebraic  signs  of  a  fraction.  In  every  fraction  there 
are  three  signs  to  consider ;  the  sign  before  the  fraction,  the 
sign  of  the  numerator,  and  the  sign  of  the  denominator. 

Thus,  +-rk'-'   ~+7'   ~^'  ^^^'     ^hen  a  fraction  is  standing 

alone,  the  sign  +  may  be  omitted  ;  as  — ^,   ^,   etc. 

—  7  o 

Since  the  names,  numerator,  denominator  and  fraction,  are 
merely  other  names  for  dividend,  divisor  and  quotient,  respect- 
ively, the  laws  of  signs  for  division  must  hold  for  fractions. 
From  division  we  have  the  following  principles : 

I.  If  the  signs  of  the  numerator  and  denominator  of  a  frac- 
tion are  both  changed,  the  sign  of  the  fraction  is  unchanged. 

II.  If  the  sign  of  either  the  numerator  or  denominator  is 
changed,  the  sign  of  the  fraction  is  changed. 

+  2  —2  —2  -1-2 

Thus,  ■— o   and  -^  are  both  positive  ;    370  and   30   are  both 

negative. 

Since  the  sign  before  a  fraction  may  always  be  considered  as 
indicating  whether  it  is  to  be  added  or  subtracted,  and  since 

139 


140  ALGEBRA 

the  subtraction  of  a  term  is  the  same  as  the  addition  of  the 
term  with  its  sign  changed ;  the  sign  before  a  fraction  may  be 
changed  if  the  sign  of  either  its  numerator  or  denominator  is 
changed. 

Therefore^  any  two  of  the  three  signs  of  a  fractioii  may  he 
changed  without  affecting  the  expressio?i  as  a  whole. 

Thus,  j.=zrh~ h~~~ '—b'   ^^^^  being  positive  ; 


and 


-a       a  a         —a  ,   ,    • 

■jr  =^=  —  j  =  —  -ziy   ^^^"  bemg  negative. 


If  the  numerator  or  denominator  be  a  polynomial,  by  §  40 
its  sign  will  be  changed  by  changing  the  sign  of  each  of  its 
terms. 


Thus, 


■x^-\-3x—l      'of-Sx+l  x^—3x+l 


103.  Multiplying  or  dividing  both  terms  of  a  fraction  by  the 
same  expression  does  not  change  the  value  of  the  fraction. 

For,  since    -=1,  by  §  54  we  get 

a _a  X _ax      ,       .     a _ax         ax     a 
l)-\x~b^'  ^^'^^  '^'  l}~~b^'  ^"^  Tx~b' 

Example  1.    Multiplying  both  terms  of —77-  by  a  +  &,  we  get 

a—b_  a^—b^ 

a+b~{a  +  bf 

'L2a^b^c 
Example  2.    Dividing  both  terms  of  10^41^2^2  by  6aWc,  we  get 

12aWc  _  26^ 
18a*b'c'  ~  3ac 

The  processes  of  multiplying  and  of  dividing  both  terms  of 
a  fraction  by  the  same  expression  are  called  reducing  the  frac- 
tion to  higher  tsrms  and  reducing  the  fraction  to  lower  terms, 
respectively. 


Example  2.    Reduce  — — 13^2  to  its  lowest  terms. 


FRACTIONS  141 

104.  Reducing  a  fraction  to  its  lowest  terms.  A  fraction  is 
said  to  be  in  its  lowest  terms  when  its  numerator  and  denom- 
inator have  no  common  factor. 

To  reduce  a  fraction  to  its  longest  terms,  divide  both  its 
numerator  and  denominator  by  all  of  their  common  factors^  or 
by  their  highest  common  factor. 

This  follows  directly  from  the  definition  of  highest  common 
factor,  §  99,  and  from  §  103. 

Example  1.    Reduce  ^a^^Lyi  to  its  lowest  terms. 

Dividing  the  numerator  and  denominator  by  their  highest 
common  factor,  4,ix?yz, 

S6x*yz^~9xz' 

ocfy^z^ 
■x*y*z'^ 

Changing  both  negative  signs  to  positive,  and  dividing  both 
terms  of  the  fraction  by  their  highest  common  factor,  x*^V, 

Qc^y^z^  _oc^y^z^  _x^  '     - 

~  —x*y*z^  ~x*y*z^  ~  y^' 

x^  -\-  2x 3 

Example  3.    Reduce  '  .,    ^ r.  to  its  lowest  terms. 

Factoring,  and  dividing  both  terms  by  their  highest  common 
laetor,  a; +  3, 

x^-{-2x-S_ix-l)(x-\-S)_x-l 

x''  +  5x+6~{x  +  2){x  +  S)~x+2' 

1  -i^^ 
Example  4.     Reduce    ,    „ =  to  its  lowst  terms. 

Changing  signs,  and  writing  both  terms  of  the  fraction  in 
descending  powers  of  a?, 

1-^     ^_     af-1     ^_ix+l)(x-l)__x-l_l-x 
x''  +  8x  +  7~     x'^  +  Scc  +  T"     {x+l)lx  +  7)        x  +  7    x-\-7' 
Note. — The  process  of  dividing  the  numerator  and  denominator  by  a 
common  factor  is  sometimes  called  cancellation. 


142 


ALGEBRA 


EXERCISE  46. 


Reduce  to  lowest  terms : 


^ 


-SaW 


6. 


8. 


-256a^^V^ 


—  4:9x^1/^  w*' 


3a.y-2/ 


28 


a;^-9a;  +  20 


i 


10  +  3£c-£c^' 

29.  x^-h2x'y-2xf-y'' 
x'—'6x^y—2xy'^^\.]f' 


30 


}f-2hc-\-&-a} 


31. 
32. 
33. 


?yr 


10m  4  16 


m'^  +  m— /2 
ac—bc—ad^-bd 


ac^ad—bc  —  bd' 

x'^x'.-^x-^ 

x'-4x'  +  2x-V^' 


105.  Reduction  of  a  fraction  to  an  integral  or  mixed  expression. 

A  fraction  whose  numerator  is  of  a  degree  equal  to,  or 
higher  than,  the  degree  of  the  denominator  may  be  reduced  to 
an  integral  or  mixed  expression  by  division. 


FRACTIONS  143 

Example  1.     Reduce ^ — -—  to  a  mixed  expression. 

B    S  56  Sx'-Qx  +  2_3x^_6x    2 

oX  oX       oX      oX 

=x-2  +  §~. 
Sx 

Example  2.    Reduce to  an  integral  expression. 

Performing  the  indicated  division, 


X 


-d^^x-\-l. 


Example  3.     Reduce ^  to  a  mixed  expression. 

Since  the  denominator  will  divide  a^  +  l,  add  and  subtract  1  in 
the'  numerator. 

Then,  ^=^^!±i=2=a!±l__2^=„,_„  +  l.      S 


a  +1        a+1        a  +1     a+1  a+1 

Example  4.     Reduce  — ^r^^^  to  a  mixed  expression. 

x—2  ^ 

Dividing,  we  get  the  quotient  Sor+l,  and  the  remainder  8. 

Hence,  3^JzS|+6=3x+l+    « 

'  X—2  x—2 

The  division  has  the  effect  of  breaking  the  numerator  into  two 
parts,  such  as  in  the  preceding  examples,  one  part  of  which  is 
divisible  by  the  denominator,  and  the  other  part  not. 


EXERCISE  47. 

Reduce  to  an  integral  or  a  mixed  expression : 

x^     '  /        '  a;  +1*  '  X  —y' 

2a;'  +  4a;4-l        ^       -    a;^  +  3a;^ 4- 3a;  + 1  a;^-3a;'>  +  2a; 


ALGEBRA 

4x'  +  2x'-Sx  +  l      Q    x'  +  lQ 
2x'-l          '       '     x  +  'I' 

11. 
12. 

x*  +  U 

X  -2  " 

x'  +  y'                             9a'  +  Sab  +  4:b' 
x  +  y'                     ■^"*         Sa-2b       ' 

x'-y' 

x-^y  ' 

144 


8. 

106.  Fractions  reduced  to  their  lowest  common  denominator. 

Fractions  may  be  reduced  to  equivalent  fractions  having  a 
common,  or  the  same,  denominator  by  §  103.  Since  the  lowest 
common  denominator  must  be  obtained  by  multiplying  each 
denominator  by  some  expression,  therefore  the  lowest  common 
denominator  must  be  the  lowest  common  multiple  of  the 
denominators. 

The  expression  by  Avhich  any  one  denominator  must  be 
multiplied  to  obtain  the  common  denominator  must  be  the 
quotient  obtained  by  dividing  the  common  denominator  by  the 
given  denominator  of  the  fraction.     Hence  the  rule : 

To  reduce  two  or  more  fractions  to  equivalent  fractions  having 
the  lowest  common  denominator,  first  find  the  lowest  common 
multiple  of  all  of  the  denominators  ;  divide  this  hy  the  denomina- 
tor of  each  fraction  in  turn,  and  multiijly  both  terms  of  the 
corresponding  fractions  by  the  quotients  thus  obtained. 

Example  1.    Eeduce  -t'   #,  and   —   to    equivalent    fractions 
having  the  lowest  common  denominator. 
The  lowest  common  multiple  of  ab,  be,  ac,  is  abc.     Dividing 

OC  CSC 

this  by  ab  gives  c.     Multiplying  the  terms  of  -v  by  c  gives  —r- . 

Dividing  abc  by  be  gives  a.     Multiplying  the  terms  of  ^  by   a 

au 
gives  -#•   Dividing  abc  by  ac  gives  b.     Multiplying  both  terms 

of  —  by  &  gives  —^.    Hence  the  required  fractions  are 

ex     ay      bz 
abc    abc    abc 


FRACTIONS  145 

Example  2.    Reduce  ^2_|.4^^3^   s^^'   ^^^  ^Hs  *o  equivalent 
fractions  having  the  lowest  common  denominator. 

x+2 x  +  2  a?— 1  x—1  X          X 

ie  +  4a;+3"'(it'+l)(a?+3)'  a?''^-9~(£i?  +  3)(x— 3/  x—?r'x^Z 

Hence  the  lowest  common  denominator  is  (a?  +  3)(u7— 3)(a7  +  l). 

Dividing  by  (ic+l)(x+3)  gives  ic— 3.     Multiplying  the  terms  of 

.T  +  2         ,        _       .  (a;  +  2)(a;-3)  ix^-x-^ 

(^•+l)(x  +  3)  '^y  ^    '^  gi^'^^  (ic+l)(a!+3)(£C-3)'  ^^af*  +  ar^-9x-9* 

Similarly,        x-1       _      (..-l)(x-fl)      _        0.^-1         . 


and 


(;:c  +  3)(x-3)     (x  +  l)(ic  +  3)(if-3)     ar^  +  x^-Ox-Q' 

x    _     a;(a?  +  3)(.r+l)      __^+JteM^^^ 
a?-3~(^+T)(x+3)(a?— 3)"~xM^'-9a?-9* 


EXERCISE  48. 

Reduce  to  equivalent  fractions  having  the  lowest  common 
denominator : 


2       15  «    2a^      W 


^'  Zx'    'Ix''    Qx  ^'  Wc'    Ic^c'    Act'b'' 

^'  W    6^^'    ia^'  ^*  xy'    yz'    xz 


3        5        1  y    j^      z       X      y 

\x^ 

5. 


6, 


£C+1'     X-V     X^-\ 

3  5 


a;^  +  3a;  +  2'    2a;^  +  5a;  +  2'    2a;='  +  3a;  +  l 


2cc  +  l  \—x  X 


10 


(^+iy'  (^+iy'  («;+ir     ^/  X  '^ 


2  a;  2a;^  ^^^^_ 


146  ALGEBRA 


iQ  ^       •'^  ^y 


J 


y'  '-^y—y^'  x'—y^' 
ii   _^    y__         ^'  y^ 

x-y'    2y-'lx'    ^{x'-yy    ^y' -  x')' 

12.  _^,    y    -1-. 

a— a     y—o     z—G 


2a  4:c 


6«-6' 
1 


13. 

^14. 

^^'  (c«-5)(a=^'    {b-c)(b^^y    (c-a)(c^' 
•|/-g    y-\-z  z  +  x  x  +  y 

^     '  (y-^)(^-»^)'  (y-^)(y-^)'  (^-^K^-y)' 

107.  Addition  and  subtraction  of  fractions.    By  §  56, 

a+b+c_a     b     c 

X  XXX 

Hence,  °+*  +  f  =^±*±£.. 

jr  *  jr     JT  >r 

Therefore^  fractions  hamng  a  common  denominator  may  be 
added  by  adding  the  numerators  for  the  numerator  of  the  sum^ 
and  using  the  common  denominator  for  the  denominator  of  the 
sum.  A  fraction  may  be  subtracted  by  chayiging  its  sign  and 
then  proceeding  as  in  addition. 

Fractions  which  are  to  be  added  or  subtracted  must  first  be 
reduced  to  equivalent  fractions  having  a  common  denominator. 

As  was  pointed  out  in  §  5,  if  either  the  numerator  or  denom- 
inator of  a  fraction  is  a  polynomial,  the  dividing  line  also 
serves  as  a  sign  of  grouping.  Consequentlj^,  in  such  cases,  the 
signs  of  the  terms  in  the  numerator  of  a  fraction  which  is 


FRACTIONS  147 

preceded  by  a  negative  sign  must  all  be  changed  when  the 
numerators  are  added. 

Example  1.     Simplify  ^J+|^+^. 

Reducing  to  a  common  denominator, 

x+1      3      iX^-l__6x'  +  Qx    9x'     2x'-2 
of      2x      SX"  Qx'         iixr'       ijx' 

_ex''  +  Qx  +  dx^ +  2x^-2 

ijx' 
_17x'  +  ex-2 

Example  2.    SimpHfy  a^Ja+^-a^-L+3-a^-L^2 

Here  ^  3  1         _ 

'  a^— 5a  +  6     a^— 4a  +  3     a^—i^a  +  2 

2  3  1 


(a-3)(a-2)     (a^3)(a-l)     {a-2){a-'i) 

2a-2  3a-6  a-3 


(a-3)(a-2)(a-l)     (a-3)(a-2)(a-l)     (a-3)(a-2)(a-l) 
2a— g— 3a  +  6-2  +  3 
(a-3)(a-2)(a-l) 
7-2a 


(a-3)(a-2)(a-l)" 


Examples.     By  addition  reduce  x  +  y  +  ~r-^.  to  a  fractional 

form. 

Here  the  integral  part  may  be  considered  as  a  fraction  whose 
denominator  is  1. 

Hence,  i»+2/  +  -^=^^  + 


x—y       1       x—y 
_x^-y^^    y^ 
x-y     x-y 

x—y 

^ 


148  ALGEBRA 

Note. — It  is  best  first  to  arrange  the  denominators  of  all  fractions 
according  to  the  powers  of  some  letter,  making  use  of  §  102  if  necessary. 


Example  4.     Simplify  >-l_  +  _^-^— -J- 
^     "^    x—1      1  —  X^     1  +  '. 


+  x 

Arranging  all   denominators  in  descending  powers  of  x,  and 
changing  signs  in  the  second  fraction,  we  have, 

1  a-  1  1  cc  1 


1     1—x^     1  +  x    x—1    x'^—l    x  +  1 
_x  4-1        X        x  — 1 
~xi^—l~x^—l~~x^  —  l 

_x  +  \—x—x-\-l 
~         x'—l 
2  —X 


V 


EXERCISE  49. 

Simplify : 

1.  h^-^'+'  ■           9.  1  +  ^. 

X           X^       ^X  X 

2.  y-H 5.  10.    X-\-zr-7—' 

DC     ac     ab  1  +  a; 

_    a;4-l  ,   1    ,2^-1  ^^        ,  ^        x^ 

3.  —-2-  +-r-:+  -.A  ■>  •  11.  a?  +  l- 


x^        hx      lOa?'^  '  x  —  \ 

4.  1  +  i  +  l.  'l%l-.'    ^^^ 


£«?/     yz     xz  \Zy  \^rx     \—x 


u. 

x-\ 

a;  +  l' 

^  6. 

3 

2         4 

a^  — £C 

ic^+aj     iK' 

7. 

2a;+l 

a;  +  2      1 

a^-2 

2£C-1       X 

8. 

3a 

^  1  ^< 

a          ^ax 

14. 
15. 


a-\-b     a  —  b 
a—b     a  +  b' 

1  1 


16.  o:-2 


2aj^  +  7£c-4     3ic^  +  13a;+4 


FRACTIONS  149 

17  _« L L.  21    -^ ^-1 

18  ^.+JiL_J^.  V    22.^^ 1       I     2y 

\-\-x     x—\     1—x^  x^-\-xi/      ^—xy      x^—if-' 

x^    'A—y        2/  —4        ^  +  2/  x-\-A  x—i 

25  1  1 


26. 


ab—ac  —  b'^  +  be      bc  —  ab—&-\^aG 

3a  +  l     2^>-l     46—1     6(^  +  1 
12a  m    ^   16c         24c?  * 


27.  .      ,A,      .+  1  1 


{a—b){b—c)     {b  —  a)(a  —  c)     {c—a){c—b)' 
'*0'  ;;;3:;,— ;:r3:7,+;:;23:::72-  V«>"'  7777-7;;.+ or^-::.+i;F 


£c— 2/     aJ  +  2/     x^-Vy'^  ^      '  2b— x     2b-]-x     x^—A^b 

OQ    _^^ 2a;  1  2a;  So; 

'*^-  a;=^-l      a;^  +  a;  +  l"^a;-l'         *^^-  ^     ^^a;"^-l      x'-^V 

32.-^-^+.    1  1 


a;— y     x  +  y     x—'ly     a;  +  2y 

Note. — In  certain  cases,  like  this  last  exercise,  it  is  best  to  add  only 
certain  ones  of  the  fractions  at  a  time.  It  saves  long  multiplications. 
Here,  add  the  first  and  second  fractions  ;  next  add  the  third  j)nd 
fourth  ;  then  add  the  sums  thus  obtained. 

33.4-,+4-o-A-^o.     34.  A— ^.4     ^  ^ 


cc  +  1     £C  +  3     x—\     a?— 3*  *  a  —  b     a^b     c—d    c-\-o 

2,3  2  3 


35. 


a;— 5     a;  +  l     a;  +  5     a;— 1* 


i  -    1 


36.  - 


a;'-^  — 1      a;''^  +  l      x^—a^     x^  +  a^ 
a;*  — 1     x*-\-x^     x*—x^  d'—b^ 


150  ALGEBRA 

39.  ^^~-x\  41.  a+     ^,_^,     +k 

■  ^    x'-hy'  ,         x'-7/  ^^-    S  +  2x     2^Sx  ,   16.x— jc^ 

40.  -+x ~.  42.  -o —-on 2 — A~' 

x—y  x-\-y  2—x       2  +  £c         £c^— 4 

108.  Multiplication  of  fractions.  Fractions  may  be  multiplied 
by  the  rule  established  in  §  54.  In  case  the  given  fractions  are 
such  that  their  product  may  be  reduced  to  lower  terms, 
the  process  of  multiplication  and  reduction  may  be  shortened 
by  first  cancelling  any  factor  of  any  numerator  hy  an  equal 
factor  of  any  deno7ninator. 

This  is  evident,  since  sucli  factors  may  be  cancelled  after  the 
multiplication.     See  §  104,  note. 

Example  1.    Find  the  product  of  i|^,   i^,   ^ 
12a^  Uab  5bc    'l2a^Uab-5bc 


W    9a-c-*  2a~"  76'^-9aV-2a 
840a362c 


\2^a?b''& 


Reducing,  =g^ 

Example  2.     Simplify   (   ^~^  ) 


x^  +  2a?, 

/  ^^-4 \  /  .T='-9\  __{x-2){x\2)     (x  +  3)(a;-3) 
\x^-^x)\x'^2xj~     x{x-Z)      ^     x{x-\-2) 

...                       „     ^             ix-2){x  +  3)   ^^,x'  +  x-e 
Cancellmg  common  factors,   = -^ -,  oi  — ^^ 

Note. — The  cancellation  may  be  indicated  by  drawing  lines  through 
the  common  factors. 

{x-2)Cs»^     (x  +  ^){s»^^     (x-2)ix  +  :^)    x'-^x-6 
-^^^^^'      xi,^^  a-iae-t^-  ~        x-x        ~      x' 

o      a-       ^^f     a'-lSa  +  SO^    a'-6a-7   ^a  +  5 
Example  3.     Simplify   ^..g^.go  x  a^-i5a  +  56  ^  ^=:i* 


FRACTIONS  151 

a'^-Sa-SO  'a='-15a  +  56'a-l~  (ii^(a,;s*4tJ)  (^^^(^.*^  (a-1) 

g  +  l 
-a-l' 

109.  In  multiplication,  any  mixed  expression  should  first 
be  reduced  to  a  fraction.  Expressions  free  of  fractions  may 
l)e  treated  as  fractions  with  the  denominator  1. 

EXAMPLE  1.     Simplify    (.  ^£^)  (y-  ^ 

\     i^-yj\     ^+yj    \3c-yJ\3c+yj 

x^—y"^' 

Example  2.     Multiply  ^.-^^^^  by  a^^  4- 4x- 21. 

x^2  x+2  (a?  +  7)(^-3) 

■x^  +  12x-v^h^'^  '^         ^^^-{x+^){x  +  7)'^  1 

(x+2)(x—2>)        x^—x—6 
x  +  5        '  ^'      x  +  5 

Note. — It  is  clear  that  the  product  of  a  fraction  and  an  integral  ex- 
pression, as  in  tlie  preceding  example,  may  be  obtained  by  merely 
multiplying  the  numerator  of  the  fraction  by  the  integral  expressioiij 
and  placing  the  product  above  the  denominator.     Thus,  in  general, 

a         an 


EXERCISE  50. 

Simplify : 


4,w2 


152  ALGEBRA 


x—y     x^  Ay 

20.-1     2y  +  3y  +  3 
^'  7+3  ^"^^^^2a;-l* 


J 


-  (-S)(.-^)- 


J 


te^-l     a;''  +  3a;4  2 
X  4-2^a;'  +  2a3  +  l' 

x^-1 


-^X 


,2         4y2 


9.  ^Z^  x^ - 

'  £C  +  2y      x^—y^ 

a  +  3  ^   a  +  4  * 

11  ^'-y'v  ^'~y' 

19.   (a; 


i 


14 
15 
16 


■( 


ab 


^     b' 


y 


a'-^b 

d^_         _ 
y'      ^)\  3£c 

_27a^  /     1 


L+iV 


J 


20 


xy—y 
'x  +  y 
x^—i/ 


b-Sa 

18.  ^a^-b^^(^^^^y 
({x±yY\ 


r^X 


21 


23 


X        a?^+y^ 


i 


/a^-hab\  /    g ^\ 

22    /^     yy.      a.^  +  y^\ 

Vy    a;A     (^+y)7' 

/x^—y'^—z^  —  2yz 
\     x^—xy—x: 


2z 


x  +  y+z 


) 


24.  (^^-a^y +  2/0x5 


25. 
26 
27 
28 


2a;+l 


X, 


^^=T6^2^M^5^+2 

.2V       /  4^2 

5c 


1.   (3a+jy3. 


/x'-Sx+   2\  /a;^-7a;  +  12\  /a;-^-5a;^ 
V£c'^-8x  +  15y  \x'-bx+  4/  V  a;-^-4  / 


'•    (a? 


x^—4x 


bx  +  QjW  +  2x 


Sx 


m- 


FRACTIONS  153 

Multiply  the  following  by  such  expressions  as  will  make  the 
products  integral  expressions : 

Suggestion  :  Multiply  by  the  lowest  common  denominator. 
a  1 


30. 
31. 


ci'-b'     a  +  b' 
2ic  +  l  ,       3 


x"^  —  1     x^  —  1     x^  +  x+1 


32.  ^— ^  + 


110.  Division  of  Fractions. 

Since  the  divide7id  =  quotient  X  divisor, 
.-  a;     a 

y    ^ 

then  -=<?Xr 

Multiplying  by  J    ^x|=^x'^X^^  (^c«.  3) 

.  =0',  since  _X-=1. 

Therefore,  ir_=_?=^X-.  (-4.^.  7) 

y     b     y     a  ^ 

Note. — A  fraction  is  said  to  be  inverted  when  its  numerator  and 

denominator  are  interchanged,  thus  j  inverted  becomes  -. 

We  can  state  the  above  formula  as  a  rule  as  follows : 
To  divide  hy  a  fraction  multiply  the  dividend  by  the  divisor 
inverted. 


154  ALGEBRA 

Example  1.     Divide  g^^,  by  3^- 

_4a'b 
-  7xy ' 

x^—1 
Example  2.     Divide  ^vi  hj  oc^  +  x  +  1. 

Considering  the  divisor  as  a  fraction  whose  denominator  is  1, 
we  have 

af—1     ,  „  ^,     x?  —  !    x^^-x-^-l 


Simplify : 


"  1. 


'-^■^+1  •       1 

{x-V){x^-^x^\) 

1 

x^^V 

X-  +  X+1 

x-\ 

-x'W 

3XERCISE    51. 

^    3    ''^''  . 

Sa'  +  Sb' 

^'  ar-h'^ 

a  +  b 

x-^-f 

x  +  y 

^'  x^-9y'^ 

'  x  +  Sy 

20x^y^       4taxy 

x'  +  Sxy  +  2y'     2x'  +  5a;y  +  3y^ 
^*  ^M^2ajy  -  Sy''^  x'  +  5£c//  +  Qy' 


7. 


3^  — 5a;— 50    *    x^  —  Qx—7 


\|      a^'— a'— 2a  ^a^— 4a-5  •       «-5 

9'  ^S^"-(-^+2--s5)- 


FRACTIONS  155 


.0.  ('S--)*(-^.' 

111.  A  complex  fraction  is  a  fraction  whose   numerator,  or 
denominator,  or  both,  involve  fractions. 

a  x  +  y 


J)  CJCy  Ql' 11 

Thus,  — '  TTT','  "TT" '  ^I'G  complex  fractions. 
d     xy 

Since  any  fraction  is  an  indicated  quotient,  a  complex  frac- 
tion may  be  simplified  by  performing  the  division  indicated. 
This  may  also  be  accomplished  by  multiplying  both  terms  of 
the  fraction  by  such  an  expression  as  will  make  tliem  both 
integral,  i.  6.,  by  their  lowest  common  denominator. 


Example  1.    Simplify  —^- 


a 
a—b 


a       a'-b' 

a  +  b 

-a-b""     o}    - 

a 

x  +  y 

X 

a 


EXAMPLE  2.     Simplify 

^ y 

y 

Multiplying  both  terms  by  xy,  their  lowest  common  denomina- 
tor, we  have 

x  +  y 

X    _xy  +  y^ 
x—y~~x^—xy' 


156  ALGEBRA 


Example  3.    Simplify  j — ^. 


1      1     y—x 


X  ^,2         ,^2 


xy 


1      1      y'^—x^      xy      y^—x^    y  +  x 


EXERCISE  52. 

Simplify : 

X  a'-b'  ^  ,      6 

/       —  ■ J         a—l-i p. 


1.  |2-  5.  — ^.  9. 


i+- 


4.  -T — 


«^-6^ 

6. 

a-b' 

b 

V)  6. 

2 

»'+2  +  ,+, 

x-y     x-\-y 
x+y     x—y 

7. 

x-y    x-Vy 

x-Vy     x-y 

a^-\-b^ 

8. 

a'-b' 

a*-aW+b' 

a-2 


-  1 


-  ^            aj+2  +  -47>                   -2^  +  1 

1  x—yx-\-y                 9x^  —  4y^ 

o.   5—  7.   ; 11.   "o Ti — • 

1  «             ^~y     ^  +  2/                    6x—ly 


X  ^"t-y    ^"~y  x^—y 

'12.  7- 


l+a3^^  a'^-a'h'^b'  ^    x 

Suggestion  :  Begin  with  the  last  complex  fraction  and  simplify 
step  by  step.    A  fraction  of  this  kind  is  called  a  continued  fraction. 


13.^ 14. 

a ^  x-V 


a  +  A-  1  +  '' 


a-l  "  '  3-a; 


FRACTIONS  157 

EXERCISES  FOR  REVIEW  (III). 

1.  What  is  meant  by  the  factors  of  an  expression?  Illus- 
trate. 

2.  Name  the  typefomis  that  have  been  factored  in  this  book. 

3.  Of  what  type  form  are  the  following?   Factor  them. 

/    {a)  Qx'-^x'y^'ixxf.  (d)  a'b-a'b'  +  aW-ab'. 

,{b)  ^a'b-'iaW  +  d'b.  ^<e)   x^'+Y'^  +  ^Y'' 

(c)  2x'-Qx\  (f)cf+''b'  +  a''b'+'. 

4.  Of  what  general  type  are  the  following?    Factor  them. 
y  (a)  xY-1.  (d)  a'  +  4ab-c'  +  4b\ 

(b)  lx'-9xy\  /(e)   x'-lQ. 

(c)  {a-by-^c\  ^(f)l  +  4xY-x*-4y\ 

5.  What  types  are  here  represented  ?    Factor  them. 
V     (a)  x'-Q4.  (c)    l+x\ 

(b)  a«  +  729.  ./   (d)  {x-iy-y\ 

6.  What  types  are  represented  in  the  following?  What  is 
the  method  used  ?     Factor  them. 

t   (a)  x'-x-SO.  ^  {e)   4a;^-12a-y  + V- 

{b)   ^x^-^x-%  (/)  15  +  2x-a;^ 

/(c)   3-2a;-8a;^  ^{g)  a«  +  a*  +  l. 

{d)  9xSj-\'hx'-2y\  (h)  b'  +  2ab-Sa\ 

7.  By  what  general  method  of  attack  do  you  factor  a  poly- 
nomial of  more  than  three  terms  ?  Illustrate  it  in  factoring 
the  following : 

\   (a)  a^  +  a'^b—a—b. 

(b)  x-\-x*—ax^—a. 

(c)  x''-a'-Qx-b'  +  9-{-2ab. 

yXd)  a^x  +  a^  +  ab-{-  acx  +  abx  -{-ac  +  bc  +  bcx. 


158  ALGEBRA 

8.  What  is  the  remainder  theorem  f     Give  the   proof.     Use 
it  in  factoring  the  following : 

^  (a)   x'-'lx^-^x-^^.  (c)   a^^-10a^4-31a-30. 

ih)    2.r^  +  lla.''  +  10a^-8.  y{d)  W^W^^^h-^Vl. 

9.  Name  the  type  form  of  each  of  the  following,  then  factor : 
V  {a)  4aV  +  6aV  — 2«a7y.  (^)   A:Q^ — \xy  ^  y"^ . 

(b)  5aV  +  10«V+5«V.  (d)  a^y  +  32a;y  +  256. 
£e)   4x*— 60?/m£c'  +  225mV. 

^(/)  4£c^-225y  ;  l-196£cy. 

(^)  x^-\-x^y  +  xi/^  +  ]/.  Jl2^)  iey— 24£c?/^  +  143^^ 

^(A)  a^^  +  2a;-£cy-2y.  {q)  a;^— 29^^  +  120. 

(0    d'-a'b  +  ab'-l)\  (r)   xY  +  2xy-V20. 

^  {j)  6y^-21m^w-8m+28m^  V  (s)   a;=*  +  a;='  +  a;  +  l. 

{k)  £c^-2£c?/  +  2/^-4.  (0    a^-a^b-Vab'-W. 

{I)    a?-2ab-\-b''-2^.  ^(u)  a'  +  b'  +  2ab-4a'b\ 

^  (m)  9c^  - 1  +  ^^^  -1-  6c^.  (y )  ay  +  «''»;''  -  ^>''a3^ - %^ 

(^0  25£c'  +  70a;?/2  +  49yV.  Ao)  2a;'  +  5£c-12. 

(o)  £cy  +  12a7y='  +  27.  (a;)  ac-bd-ad+bc. 


10. , Factor: 
"^{a)   9«« 


6a^  +  l.  /O*)  Qx'-lSxyi-Qy\ 

(b)   12a'  +  7ab-10b\  ^  (k)  7x'-h2bxy-12f, 

Uc)  a'-^a'b'  +  4:b\  {I)    16«'^-48a  +  35. 

*  {d)  4a;*  -  4a;V'  +  9y*.  (/?i)  a"'^  -  ^'". 

{e)  IQa'  +  lQab-W.  '' {n)  a'  +  216. 

(^)  a;^«  +  2a;«  +  l.  (/>)  a'  +  a'  +  l. 

(h)  a'-b'-a-b.  (q)  Sx'y  +  Sxy-}-SxY. 

(^)    x*  +  xy+y\  (r)  a^—a—a'^b—ab. 


f^RACTIONS  150 

(s)  l  +  a—b—ab.  (v)  ahx^-\^{a'^^-h-)xy-\-aby'^. 

(t)    4:a'b'-(a'  +  b'-cy.  (w)  a'b'  +  a'x'-b'x'-x*. 

(u)  Sbc-4ad+Qac-2bd.  (x)  Qa'  +  9ab-3b-2a. 

11.  Find  the  11.  C.  F.  of : 

(a)  x'-S,  x'-2x'-{-x-2. 

(b)  x'-V2x  +  S^,  x'-2x'-19x  +  20. 

(d)  x'  +  Sx'  +  4x  +  12,  x'  +  4x'  +  4x  +  ^. 

12.  Find  the  L.  C.  M.  of : 

(a)  Qx\  x'-2x,  3£c='-12a;  +  12. 

(b)  2a%ba'-^ab,2ab  +  2b. 

(c)  x^—x^  —  Qx,  x^  +  2x\  x^  —  Qx^  +  dx. 

(d)  Gx'  +  7x-^,2x'-x'  +  Sx—4. 

13.  Explain  the  process  by  which  algebraic  fractions  are 
added. 

14.  What  law  of  signs  must  be  observed  when  a  fraction 
is  subtracted? 

15.  Simplify : 

/  X      3      .  2  1 

(a) 


x'-l  '  x'-Sx-4:     x'-x-2' 
(1\       9^  +  17       ,    2x-l       2x-\-l 


x'-2x-4S  '  2a;+12     2a;-16 

17.  By  what  must  a  fractional  expression  be  .multiplied  in 
order  to  obtain  an  integral  expression  ? 

18.  Multiply  by  such  an  expression  as  will  make  integral: 


1 QO  ALGEBRA 

J.       1  3 2_ 

3(^-1)       3a; 9^ 

^  ^      i«-2  a;  +  2     a;  +  l* 


19.  Sin,plify   (-^-l)^(3.,-J^) 

20.  Simplify      ^V'^'^V^    ^x'y^—x'^ 

x^-\-xy-\-y^  '    y^—x^ 


21.  Simplify       1     1     X^3ipj3- 

b     a 

22.  Divide  ^  +lz:?by  y^  -III?  and  express  the  quo- 

J.  ~T~ XX  X  "I   X  X 

tient  in  its  simplest  form. 

23.  Simplify: 

W    \^,     «;  •  \a     b)^    a  +  b  ^b\a-by 

^^     \bTc~^^rFcJ\^Tb'^^^^b~'      a'-b'      J' 

/2(^)_«+.wi    ly 

^   ^    \     a-^x      a—xj  \a     xj 

^  ^     Va;'-y=^     x+yj  \  2zy  J         x-y 

ab—b'^a^-b'' 
a^—ab        ab  1 


CHAPTER  XII. 
LINEAR  EQUATIONS— ONE  UNKNOWN  NUMBER. 

112.  We  showed  in  Chapter  II  the  meaning  of  an  equation 
and  how,  by  the  use  of  axioms,  to  solve  the  simplest  kind  of 
equations  containing  one  unknown  number.  Now  that  positive 
and  negative  numbers  and  fractions  have  been  discussed,  we 
return  to  the  discussion  of  equations. 

113.  An  equation  is  integral  with  respect  to  its  unknown 
numbers  when  both  of  its  members  are  integral  with  respect 
to  those  numbers.     Otherwise  it  is  a  fractional  equation. 

Thus,  x-\-§=Qx-\-5  and  2^^  +  3.x^=8  are  integral  equations  in  x. 
Sx^—xy  +  Si/=7  is  integral  with  respect  to  both  a?  and  y.    The 

equation  -  +  -=10  is  fractional  with  respect  to  both  x  and  y. 
X    y 

A  rational  equation  is  one  in  which  both  members  are  rational 
expressions  Avith  respect  to  the  unknown  numbers.  Otherwise 
it  is  called  irrational.  The  equation  j/a;— i/a;— 1  =  2  is  irra- 
tional. All  equations  here  to  be  considered  are  rational  equa- 
tions. 

114.  The  degree  of  a  rational  integral  equation  is  the  degree 
of  its  term  of  highest  degree  with  respect  to  the  unknown 
numbers. 

Thus,  x  +  3=4x  and  3ic— 2i/=10  are  of  the  first  degree. 

x^—5x+Q=0  and  x^—2xy  +  y=9  are  of  the  second  degree. 
a^  +  a?"— x=0  is  ot  the  third  degree. 

a?* — 1/^ = 6  is  of  the  fourth  degree. 

161 


162  ALGEBRA 

An  equation  of  the  first  degree  is  also  called  a  linear  equation.* 
One  of  the  second  degree  is  called  a  quadratic  equation. 
One  of  the  third  degree  is  called  a  cubic  equation. 
One  of  the  fourth  degree  is  called  a  biquadratic  equation. 

115.  Equivalent  equations.  Equations  which  have  the  same 
solutions  are  called  equivalent  equations. 

Thus,  6a?— l==4a?  +  7  and  2a?=7+l  are  equivalent,  for  each  is 
satisfied  by  a?=4,  and  by  no  other  solution. 

In  Chapter  II  linear  equations  were  solved  by  the  use  of 
axioms. 

Thus  to  solve                                   6x— 5=4a?  +  l.  (1) 

Adding  5  to  each  member,                     6x=4:X  +  6.  Axiom  1  (2) 

Subtracting  4x  from  each  member,     2x=6.  Axiom  2  (3) 

Dividing  each  member  by  2,                   x=3.  Axiom  4  (4) 

It  will  be  seen  that  this  work  consists  of  deducing  in  the 
successive  steps,  successive  equations,  each  one  equwalent  to 
the  preceding  one.  Thus,  equations  (1),  (2),  (3),  (4),  in  the 
above  example  are  all  satisfied  by  a3  =  3,  and  by  no  other  value 
of  X  ;  hence  they  are  all  equivalent.  Therefore,  the  solution  of 
the  last  equation  is  the  required  solution  of  the  given  equation. 

116.  Observe  that  to  change  the  given  equation  of  §  115  to 
an  equivalent  one  whose  first  term  consisted  of  a  multiple  of 
the  unknown  number,  and  whose  second  term  was  a  known 
number,  i.  e.  to  reduce  the  given  equation  to  the  form  ax  =  b, 
we  proceeded  as  follows  : 

I.  The  known  member  in  the  first  member^  with  its  sign 
changed^  was  added  to  both  members  to  free  the  first  member 
from  known  numbers. 

*  The  name  linear  equation  is  derived  from  the  fact  that  the  equation 
of  the  first  degree  with  two  unknown  numbers  has  a  pecuhar  relation 
to  a  straight  line.     See  §  134. 


LINEAR  EQUATIONS— ONE  UNKNOWN  NUMBER        163 

II.  The  term  iiwolmng  an  unknown  number  in  the  second 
member  urns  added  vnth  its  sign  changed  to  both  members  to  free 
the  second  member  of  unknown  numbers. 

This  process  gave  a  new  equation  in  which  certain  terms 
that  appeared  in  a  member  of  the  old  equation  appeared  in  the 
opposite  member  of  the  new  equation  with  their  signs  changed. 

Tills  result  is  equivalent  to  transferring  terms  from  one  mernber 
to  another  and  changing  the  signs  of  the  transferred  terms. 

This  process  is  called  transposition.  The  term  is  said  to  be 
transposed. 

Note.  The  pupils  should  use  the  correct  phraseology  of  -'adding 
equals  to  both  sides  of  tlie  equation,"  until  the  thing  actually  done  is 
firmly  fixed  in  mind.  When  this  is  thoroughly  understood  the  briefer 
form,  "transpose",  may  be  used  if  desired.  Care  should  be  taken^ 
however,  that  the  pupil  does  not  say  "transpose",  and  mechanically 
perform  tlie  process,  forgetting  what  he  has  really  done. 

117.  To  solve  a  linear  equation  for  one  unknown  number. 

The  following  general  method  may  be  used  in  solving  any 
linear  equation  for  one  unknown  number  : 

(1)   Remove  all  signs  of  grouping.,  if  any  exist. 

{2)  Transform,  the  given  equation  into  an  equivalent  one 
having  all  the  unknown  numbers  in  the  first  member,  and  all 
terms  free  of  the  tinknoion  number  in  the  second  member. 

(3)    Unite  like  terms. 

{J/)  Divide  both  members  by  the  coefficient  of  the  unknown 
number. 

To  check  the  transformed  equation,  see  if  the  terms  that 
were  cancelled  in  any  member  of  the  given  equation  reappear  in 
the  other  member  of  the  new  equation  with  signs  changed. 

Example  1.     Solve  7ic+15=4a?  +  3. 


164  ALGEBRA 

1.  Adding  -4a?-15,         7^-4^=3-15.  (Ax.  1) 

2.  Uniting  like  terms,  3x=— 12. 

3.  Dividing  by  3,  x=—4,  the  solution.  (Ax.  4) 

Check.     Substituting  for  a?  its  value  —4,  the  given  equation  be- 
comes —28  +  15=  — 16  +  3,  an  identity. 

Example  2.     Solve  3(a  +  l)  =  12  +  4(a-l). 

1 .  Removing  signs  of  grouping,  3a +  3=12  + 4a— 4. 

2.  Adding  -4a-3,  3a-4a=12-4-3.  (Ax.  1) 

3.  Uniting  like  terms,  — a=5. 

4.  Dividing  by  —1,  a=  — 5,  the  solution.  (Ax.  4) 

Check.     Substituting  for  a  its  value  —5,  the  given  equation 
becomes  3(-5  +  l)  =  12  +  4(-5-l), 

or  —12=— 12,  an  identity. 

Example  3.    Solve  (5— a-)(l  +  a?)  =  (2— .T)(4  +  a?). 

1.  Removing  signs  of  grouping,  ^  +  4:X—x'^=S—2x—x'^. 

2.  Adding  x^  +  2x- 5,  4x-x'  +  2x-\-x'=8-5.  (Ax.  1) 

3.  Uniting  like  terms,  6x=3, 

4.  Dividing  by  6,  ^=h  ^^^^  solution. 

(Ax.  4) 
Check.     Substituting  for  x  its  value  |,  the  given  equation  be- 
comes (5-^)(l  +  J)  =  (2-i)(4  +  |), 
or  4^1|=l|-4^,  an  identity. 

EXERCISE  53. 

Solve  the  following  equations  for  x  : 

1.  2ic  +  7  =  14-5i«.  4.  12-Ux=Q—9x. 

2.  30a;  +  24  =  60  +  48a5.  5.  8a;-7  =  3a;  +  3. 

3.  19i«-22  =  8ic-17.  6.  14aj  +  20-12= -20a;  +  35i«. 


LINEAR  EQUATIONS— ONE  UNKNOWN  NUMBER        165 

7.  2Sx-^b  =  21x-10x-U7.       9.  ^{2-4x)  =  4{l-dx). 

8.  S{x-2)=2(x-S).  10.  x-(Q-2x)  =  9(x-l). 

11.  2(a;-l)  +  3(a;-2)  +  4(ic-3)  =  0. 

12.  2x-b(20-x)-i)  =  0. 

13.  5(2a;  +  l)-7  =  3(2aj-7)  +  51. 

14.  bx-Q(S-4x)=^x-7(4  +  x). 

15.  2(a;+12)-(a;-3)=0. 

16.  (a;-2)(a3-3)  =  (a;-4)(a;-5). 

17.  S(x  +  4)(x-2)-^  =  S(x  +  b)(x-S)+x. 

18.  (a;  +  2)^-a;='=a;-5. 

Solve  the  following  equations  for  a : 

19.  l  +  (a-^y  =  (a  +  iy-4. 

20.  («-2)(a-5)  +  (a-3)(a-4)-2(a-4)(a-5). 
21.  2.5a-6.75-1.25a-3.  22.  0.75a  +  2(l-1.2«)=0. 

23.  2-6.9a(l-2a)-2(6.9a'^-3). 
24.  \a-\a  =  2.  26.  Ja  =  4-lia. 

118.  Some  fractional  equations  may  be  changed  to  equivalent 
linear  equations  by  multiplying  both  members  by  such  an  ex- 
pression as  will  destroy  all  fractions  (Axiom  3).  The  neces- 
sary multiplier  will  evidently  be  the  lowest  common  denominator 
of  all  the  fractions.  This  process  is  called  clearing  of  fractions.* 
The  pupil  should  see  that  the  real  process  is  multiplying  both 
members  by  the  same  number. 

*Like  the  term  "transpose,"  the  term  "clearing  of  fractions"  is 
often  used  by  pupils  without  their  knowing  the  real  process  involved 
and  the  authority  for  it.     A  pupil  sometimes  thinks  that  clearing  an 

equation  as  ^^-—j=- of  fractions  consists  in  multiplying  the  first 

member  by  1  +  a7  and  the  second  by  2x-\-4  rather  than  each  member  by 
(l+a;)(2a;-h4). 


56 

ALGEBRA 

Thus,  solve 

1_ 

X 

2  . 
~3x' 

5 

6' 

Multiplying  both  members  by 

6x,  6- 

-4  = 

:5X. 

Adding  —5x  +  4- 

-6, 

5x= 

=4-6. 

Uniting  terms, 

5x= 

r-2. 

Dividing  by  —5, 

X- 

=|. 

(Ax.  3) 
(Ax.  1) 

(Ax.  4) 

If  a  multiplier  be  used  whose  degree  in  the  unknov/n  number 
Is  higher  than  that  of  the  lowest  common  denominator,  the 
resulting  equation  will  usually  not  be  equivalent  to  the  given 
equation.     (See  §  173).     Hence  the  rule  : 

To  clear  an  equation  of  fractions  7nulti2^ly  all  terms  in  both 
members  by  the  lowest  common  denominator  of  all  the  fractions 
in  the  equation. 

It  should  be  noticed  in  the  following  examples  that  the  easiest 
way  to  multiply  a  fraction  by  the  lowest  common  denominator 
is  first  to  divide  the  lowest  common  denominator  by  the  denomi- 
nator of  the  fraction,  then  multiply  the  numerator  by  this  quo- 

2a 
tient.     For  example,  to  multiply  ^—  by  eer*,  we  divide  ^x"  by  3a? 

oX 

obtaining  2a?\  then  multiply  2a  by  2x^  obtaining  4aa?^,  the  product. 
If  a  fraction  is  preceded  by  the  negative  sign,  the  sign  of  every 
term  of  the  numerator  must  be  changed  when  the  multiplication 
is  performed.     See  §  5. 

Example  1.    Solve  -^,_-^=-A_ 

x—2    x+2    x^—4: 

The  lowest  common  denominator  is  a?^— 4.     Multiplying  by 

x{x  +  2)—x(x—2)=4.  (Ax.  3) 

Removing  signs  of  grouping,  x^  +  2x—x'^  +  2a?=4. 

Adding,  4a7=4. 

Dividing  by  4,  x=l,  the  solution. 

(Ax.  4) 
a?— 3    a?— 4    a?— 6     x—7 


Example  2.     Solve 


.T— 4     x—5     x—7    x—8' 


LINEAR  EQUATIONS— ONE  UNKNOWN  NUMBER        167 

Here  there  will  be  an  advantage  in  adding  the  fractions  in  each 
of  the  members  before  clearing  of  fractions. 

1-  Adding,  ^__zl_^=___l_^. 

2.  Multiplying  by  (^—4)(a?—5)(a?—7)(x— 8), 

-l{x-7)ix-8)  =  -l(x-4){x-5).  (Ax.  3) 

3.  Dividing  by  -1,  {x-7){x-8)  =  (x-4){x-5).         (Ax.  4) 

4.  Removing  signs  of  grouping, 

x''-15x+5Q=x''-9x  +  20. 

5.  Adding,  — x'^  +  9x— 56, 

x^-15x—x''  +  9x=20-5Q.  (Ax.l) 

6.  Uniting  terms,  — 6x=— 36. 

7.  Dividing  by  —6,  a?=6,  solution.  (Ax.  4) 

EXERCISE  54. 

Solve  the  following  for  x :  . 

Q         3  CC  +  1    ,        £C^ 

^'  ;T-ri-:;r—i+:;j- 


1. 

3      1 
2a5     4* 

2. 

j«  +  3     o 
«.-3-^- 

3. 

»^  +  l      2      1 
a;'    '^Sx     X 

4. 

2a^+l         1        a; 
4         05-1     2' 

5. 

1             2         3 

2x  +  l     x  +  l     'Ix 

6. 

^-2^03  +  1     ^• 

7. 

cc  +  1       a^-l       l-f2a; 
3a;^l     3.r+l     W-l' 

R 

1,1-1 

£c  +  l     a;  — 1     x^  —  1 

10.  ?Lz|  +  ?i=f=,. 
a;— 7     a;— o 

£C  — 2     05  +  2 
^2    x  —  '^x  —  t) 

/13.-A^l^-:i^^  +  7. 

£C  +  1  aJ— 1 

14.  2  +  -?^=^^. 
05  +  3     a;+7 

15    3a;  +  7_6cc-2 

4aj  +  8     8a.— 5* 

1R    8     a;-3_a3  +  l 

2a;  +  3  '  2£c-3     4a;^-9'  a5"^a;  +  3     cc-1" 


168  ALGEBRA 

17.  2a;-3     ^_  a;  +  5      11     '  '  ^  jjO.         ^^ 


2£c-4     "^     dx-Q      2*  '  aj'  +  5a;  +  6     a^^2     a3  +  3 

j^g    a?     a;^  — 5a;_2  21       ^  ^  ^  ^ 


3     3a;— 7      3*  x—^     x—1     x—4:     x—2 


X 


^g    ^ 3_    4        x-\-l  Qo       3a;  2a;    _2a;^— 5 

«„    3a;  +  5     3a;''  +  5a;— 4 


24. 


4a;-3     4a;^-3a;  +  2* 

a;— 7     a;— 9  _a?— 13     a;— 15 
a;— 9     a;  — ll~~a;— r5     a;— 17* 


OK    a;— 5  ,  a;— 7     a;— 4  ,  x 
»o.  — =4- 


26. 
27. 

28. 


a;— 7     a;— 9     a;— 6     a;— 10 

a;        a;  +  l     a;— 8     a;— 9 


a;— 2     a;— 1     a;— 6     a;— 7 
13  2 


a;  +  3     9— a;'     3— a; 

3(a;-l) 3^  _    9 

x—2       a;  +  2     a;  +  l* 


og    5(a;  +  6)       2(5a;  +  2)_^ 
a;  +  10  2a;-l 

a;— 5  7— a;  2a;  — 15 


30. 


„i        9a;  +  17       ,  2a;-   1     2a;  +  l 
^A.   -5 — rr— r-ro  + 


32. 


a;'-2a;  +  48  '  2a;  +  12     2a;-16 

5  3a;  +  5_8  +  3a; 

l-9a;^  +  3a;-l~l  +  3a;* 


119.  The  formula.  An  equation  often  contains  more  than 
one  general  number.  In  that  case  it  may  be  solved  for  the 
value  of  any  one  of  these  general  numbers.  It  is  clear  that  in 
such  cases  the  value  found  for  one  of  the  general  numbers  may 
be  an  expression  containing  the  others,  and  hence,  the  solu- 


LINEAR  EQUATIONS— ONE  UNKNOWN  NUMBER        109 

tion  may  not  be  a  single- valued  or  definite  number.     Such  an 
equation  is  sometimes  called  a  formula.* 

Thus,  ax  +  b=ac  may  be  solved  for  a,  x,  6,  or  c  ;  but  the  value 
thus  found  for  either  will  consist  of  an  expression  containing  the 
others. 

If  the  formula  be  linear  with  respect  to  a  certain  general 
number,  it  may  be  solved  for  that  number  by  the  method  of 
the  preceding  section. 

Example  1.     Solve         ax-\-b=ac    tor  x. 

1.  Adding  — &,  ax=ac—h.  (Ax.  1) 

2.  Dividing  by  a,  x=^^:^^'  (Ax.  4) 

Example  2.     Solve         ax-]rh=aG    for  &. 

1.  Adding  —  aa?,  b=ac—ax.  (Ax.  1) 

Examples.     Solve        ax  +  h=ac    fore. 

1.  Adding  —ac—ax—b,  ~ac=—b—ax.  (Ax.  1) 

2.  Dividing  by  —a,  c=~   ~ (Ax.  4) 

Or,  changing  signs,  c= (§102) 

Example  4.     Solve  ax-\-b=ac    for  a. 

1.  Adding  —ac—b,  ax—ac~—b.                                       (Ax.  1) 

2.  Uniting  terms,  a{x—c)  =  —b. 

3.  Dividing  by  x—c^  a= (Ax.  4) 

X  —  c 

Or,  changing  signs,  a=— — -•  (§  102) 

Note. — The  student  will  find  it  of  great  value  to  be  able  to  solve  a 
formula  in  taking  up  the  study  of  Geometry  and  Physics. 

*  A  formula  expresses  a  law  in  mathematical  symbols.  The  type 
forms  of  multiplication  or  division  are  really  formulas.  When  a 
formula  is  expressed  in  words  it  is  a  principle.  When  expressed  as  a 
direction  how  to  do  a  thing  it  is  a  rule.  Thus,  the  formula  of  §  110  was 
expressed  as  a  rule. 


170  ALGEBRA 

EXERCISE  55. 

1.  ^ax—h=cy.     Solve  for  x. 

2.  xy-Vx=y^~y—^.     Solve  for  x. 

3.  a{x  —  l)-{^a=x.     Solve  for  a. 

4.  «(.T  +  3)  +  5(£c-3)  =  c(£c-l).     Solve  for  h. 
^6.  {a— «;)(«  +  2£c)  =a^-\-  x^.     Solve  for  a. 

6.  («  +  5a^)  {h  +  a^c)  -  «5(a^^  + 1)  =  «'  +  ^'.    Solve  for  x. 

7.  2(2^-l)  +  3  =  «(^  +  2).     Solve  for  t. 

8.  3(^  +  «  +  a;)  +  2(^  +  a— £K)=-a7.     Solve  for  if. 
9.  Solve  Ex.  1  for  2/.  12.  Solve  Ex.  4  for  c. 

10.  Solve  Ex.  3  for  x.  13.  Solve  Ex.  7  for  a. 

11.  Solve  :Ex.  4  for  a;.  14.  Solve  Ex.  8  for  a. 

The  following  formulae  express  important  laws  in  Physics. 
lb,  s=vt.     Solve  for?;.  l^.  E=\Mv\     Solve  for  J/: 

16.  v  =  at.     Solve  for  t.  J/y2 

,      ,,      ,  20.  i^=— -.    Solve  for  r. 

17.  s=\at\     Solve  for  a.  ^' 

18.  TT^i^s.     Solve  for  i^.         21.  6^=i«(2^-l).     Solve  for  ?;. 

22.  PD=  Tr-7>'.     Solve  for  P. 


(7     i? 


ic 


23.---.    Solve  for  7?.  26.  7>=-^i-,.    Solve  for  ^^j. 

c       r  V!) — v") 

24.  ^,=^'    Solve  for  P.  27.  J^-|-(7+32.     Solve  for  C, 

25.  (7=:f.    Solve  for  P.  28.4=-+^-    Solve  tor  7?. 

11  Br      r' 

120.  Problems  solved  by  use  of  linear  equations  with  one  un- 
known. 

'  In  §  26  we  showed  how  problems  could  be  solved  by  the  use 
of  the  equation.     The  important  steps  in  the  process  of  solving 


LINEAR  EQUATIONS— ONE  UNKNOWN  NUMBER        171 

such  problems  by  use  of  the  linear  equation  with  one  unknown 
number  are  as  follows : 

(1)  Fvrst  represent  by  some,  letter  one  of  the  iinknovm  numbers 
mentioned  in  the  prohle'm. 

{2)  Then,  from  conditions  stated  in  the  problem,  form  ex- 
pressions containing  this  assumed  letter  xnhich  icill  represent  the 
values  of  any  other  luiknown  numbers  mentioned  in  the  problem. 

{3)  By  means  of  some  other  statement  in  the  problem  it  should 
then  be  possible  to  form  an  equation  betioeen  these  expressions. 

{4)   Solue  this  equation  and  interpret  the  result. 

Example  1.  The  difference  between  the  squares  of  two  con- 
secutive whole  numbers  is  121.     Find  the  numbers. 

Let  x=  the  less  number. 

Then,  it'  +  l=  the  greater  number,  for  the  numbers  are  con- 
secutive. 

Therefore,  x^  and  {x-\-  Vf  will  represent  the  squares  of  the  two 
numbers. 

Hence,  (^  +  1)^— x^=121,  for  the  difference  between  their  squares 
is  121. 

Removing  the  signs  of  grouping,  x'^-\-2x-\-l—x^=121. 

Whence,  2.r=120. 

Therefore,  a?=60,  the  less  number, 

and  07  4-1=61,  the  greater  number. 

Check.     (61)^-(60)2=3721-3600=121. 

Example  2.  The  length  of  a  room  exceeds  its  breadth  by  4 
feet;  and  if  each  had  been  increased  by  4  feet,  the  area  would 
have  been  increased  by  128  square  feet.  Find  the  dimensions  of 
the  room. 

Let  x=  number  of  feet  in  the  breadth. 

Then,.         a?  +  4=  number  of  feet  in  the  length. 

Hence,  x{x+^)=  number  of  square  feet  in  the  area.  Rule  for 
area  of  a  rectangle. 


172  ALGEBRA 

If  the  breadth  and  length  were  each  increased  by  4  feet,  they 
would  be  a:+4  and  x+8,  respectively.     And  the  area  would  be 

(x  +  4)(a?  +  8). 

From  the  condition  of  the  problem, 

(x -{-4){x  +  8)  —x{x  +  4)  =  128.     (State  the  condition 
that  gives  this  equation.) 

Removing  (    ),  x'  +  12x  +  d2-x'-4.x=128. 
Whence,  8a;=96. 

a?=12,  the  breadth. 
x+4=16,  the  length. 

Example  3.     The  sum  of  two  numbers  is  21,  and  the  quotient 
of  the  less  divided  by  the  greater  is  f .     What  are  the  numbers  ? 
Let  x=  the  less  number. 

Then  21— £c=  the  greater  number.     (Why?) 
Hence,  since  their  quotient  is  |,  we  have 

x        2 
21-i»~5' 

Clearing  of  fractions,  5x=^2—2x.      (What  multiplier  is  used  ?) 
Whence,  7x=42, 

and,  •  x=Q,  the  less  number. 

21—07=15,  the  greater  number. 

Example  4.  A  tank  can  be  filled  by  one  pipe  in  24  minutes,  by 
a  second  pipe  in  32  minutes,  and  by  a  third  pipe  in  40  minutes. 
If  all  three  pipes  run  at  once,  how  long  will  it  take  to  fill  the 
tank  ? 

Since  the  first  pipe  alone  could  fill  the  tank  in  24  minutes,  in 
one  minute  it  could  fill  ^^  of  it. 

Likewise,  in  one  minute  the  second  pipe  alone  could  fill  ^\  of  it; 
and  the  third  pipe  alone  could  fill  jV  of  it. 

Hence,  in  one  minute  the  three  together  could  fiH^V+sV  +  ^V 
of  it. 

Let  X  =  number  of  minutes  required  for  the  three  pipes  to- 
gether to  fill  the  tank.     Then  in  one  minute  they  could  fill  -  of  it. 

X 


LINEAR  EQUATIONS— ONE  UNKNOWN  NUMBER        173 

Clearing  of  fractions,  20x+ 15a? +  12ic=480.      (What    multiplier 
was  used  ?) 
Uniting  terms,  47a? =480. 

a[;=10{f,  number  of  minutes. 


EXERCISE  56. 

1.  A  has  170,  and  B  has  110.  How  much  must  A  give  to  B 
in  order  that  he  may  then  have  just  three  times  as  much  as  B? 

Suggestion. — If  x  representF-  the  required  amount  given  by  A,  what 
will  each  then  have?  What  condition  of  the  problem,  then,  gives  an 
equation  ? 

2.  Divide  50  into  two  parts  whose  difference  is  26. 

3.  Find  two  numbers  whose  sum  is  1  and  whose  difference 
is  15. 

4.  Find  two  numbers  whose  difference  is  15,  and  whose  sum 
is  f  of  their  difference. 

Suggestion. — If  x=:  the  smaller,  what  must  equal  the  larger?  What 
equation  follows  ? 

5.  Find  the  number  the  sum  of  whose  half,  third  part,  and 
fourth  part  is  26. 

6.  Find  two  numbers  whose* sum  is  36,  one  of  which  is  |  of 
the  other. 

7.  Find  the  number  such  that  \  of  it  shall  exceed  i  of  it  by  2. 

8.  Find  two  numbers  whose  sum  is  28,  and  such  that  one 
exceeds  7  times  the  other  by  4. 

Suggestion. — What  two  conditions  in  the  problem  ?  If  one  number 
is  X  and  their  sum  28,  what  must  the  other  number  be?  What  equa- 
tion follows  from  tlie  second  condition  ? 


174  ALGEBRA 

9.  Find  two  numbers  whose  sum  is  Gl  and  difference  11. 

Suggestion, — When  x=  one  number,  either  of  two  conditions  will 
give  the  other.  What  are  the  conditions?  Show  that  you  may  use 
either  condition  to  get  the  expression  for  the  second  number  and  the 
other  condition  to  get  an  equation. 

10.  What  number  increased  by  i  of  itself  and  20  is  2  more 
than  double  itself? 

11.  Eight  times  the  difference  between  one-fourth  and  one- 
third  of  a  number  is  32  less  than  the  number.  What  is  the 
number  ? 

12.  Find  the  number  that  exceeds  20  by  as  much  as  i  of  the 
number  exceeds  7. 

13.  Find  two  consecutive  Avhole  numbers  whose  sum  ex- 
ceeds 25  by  as  much  as  25  exceeds  15. 

14.  There  is  a  certain  fraction  whose  value  is  J ;  and  if  its 
numerator  Avere  greater  by  2,  and  its  denominator  less  by  2, 
its  value  Avould  be  i.     What  is  the  fraction  ? 

15.  What  number  added  to  the  numerator  and  to  the 
denominator  of  j\  will  give  a  fraction  equal  to  |  ? 

16.  John  is  six  years  older  than  James ;  and  in  five  years 
John  will  be  3  times  as  old  as  James  was  3  years  ago.  What 
are  their  ages  ? 

17.  A  father's  age  now  is  4  times  as  great  as  that  of  his  son ; 
and  4  years  ago  it  was  6  times  as  great.     What  are  their  ages  ? 

»    18.  A  horse  sold  for  $132.50,  which  was  6  %  of  the  cost 
more  than  the  cost  to  the  original  owner.     What  did  it  cost  ? 

Suggestion. — 6^  means  i^--^.  If  ic  =  the  cost,  then  -^^  a?  =  the  gain 
and  |^^a?=  the  selling  price. 

19.  A  man  invests  i  of  his  capital  at  5%  and  the  remainder 
at  6%.     Ilis  total  income  is  $4080,     What  is  his  capital ? 


LINEAR  EQUATIONS— ONE  UNKNOWN  NUMBER        175 

'  20.  16  is  changed  into  51  coins.  If  each  coin  is  either  a 
quarter  or  a  dime,  how  many  of  each  are  there  ? 

•  21.  A  train  leaves  a  station  and  travels  at  the  rate  of  40 
miles  an  hour.  Two  hours  later  a  second  train  leaves  the 
station,  and  travels  in  the  same  direction  at  the  rate  of  55 
miles  an  hour.     Where  will  the  second  train  pass  the  first  ? 

•  22.  A  tank  is  fitted  with  two  pipes.  One  can  empty  the 
tank  in  30  minutes;  the  other,  in  25  minutes.  If  the  tank 
is  two- thirds  full,  and  both  pipes  are  opened,  in  what  time 
will  it  be  emptied  ? 

<  23.  A  laborer  was  hired  for  60  days.  Each  day  that  he 
worked  he  was  to  receive  12.25  and  board  ;  and  each  day  that 
he  was  idle  he  was  to  receive  no  pay,  but  was  to  be  charged 
60  cents  for  his  board.  At  the  end  of  60  days  he  received 
1106.50.     How  many  days  did  he  work? 

,  24.  A  rectangular  field  is  6  rods  longer  than  it  is  wide  ;  and 
if  the  length  and  breadth  were  each  4  rods  more,  the  area 
would  be  120  square  rods  more  than  it  is.  Find  the  dimen- 
sions of  the  field. 

25.  What  number  mvist  be  subtracted  from  each  of  the  four 
numbers,  12,  14,  18  and  10,  so  that  the  product  of  the  first  two 
remainders  shall  equal  the  product  of  the  last  two  ? 
^^  26.  A  man  rows  down  a  stream  at  the  rate  of  5  miles  an 
hour,  and  returns  at  the  rate  of  2  miles  an  hour.  He  returns 
to  his  starting  point  in  7  hours.  At  Avhat  rate  does  the  stream 
flow?  How  far  down  the  stream  does  he  go?  How  fast  can 
he  row  in  still  water  ? 

27.  Find  a  number  such  that  i  of  it  shall  exceed  f  of  it  by  9. 

28.  B  has  $40  more  than  A,  C  has  i  as  much  as  B,  and  be- 
tween them  they  have  $360.     How  much  has  each  ? 

29.  The  difterence  between  the  squares  of  two  consecutive 
whole  numbers  is  23.     What  are  the  numbers  ? 


176  ALGEBRA 

30.  If  a  certain  number  be  added  to  8  and  to  11,  and  the 
first  sum  be  divided  by  the  second  sum,  the  quotient  will  be  |. 
What  is  the  number  ? 

31.  What  sum  at  6%  simple  interest  will  amount  to  1413  in 
3  years  ? 

32.  At  what  rate  simple  interest  will  $265  amount  to  $807.40 
in  two  years  ? 

33.  If  linen  costs  i  as  much  as  silk,  and  I  spend  $19.25  in 
buying  10  yards  of  silk  and  15  yards  of  linen,  find  the  cost  of 
each  per  yard. 

34.  A  room  is  2  feet  longer  than  it  is  wide,  and  if  its  length 
were  increased  by  4  feet  and  width  diminished  by  3  feet,  its 
area  would  not  be  changed.     What  are  its  dimensions  ? 

35.  Two  pedestrians  started  at  the  same  time  from  points 
44|  miles  apart,  one  traveling  at  the  rate  of  2i  miles  an  hour 
and  the  other  at  the  rate  of  2|  miles  an  hour.  When  and 
where  did  they  meet  ?  / 

36.  Find  the  time  between  4  and  5  o'clock  when  the  hands 
of  a  clock  are  together. 

Suggestion. — Let  x  represent  the  number  of  minute  spaces  which  the 
minute  hand  lias  traveled  from  4  o'clock  on  until  it  overtook  the  hour 

hand.     Then  j^  will  be  the  number  of  spaces  which  tlie  hour  hand 

has  traveled  meanwhile.    The  difference  is  20.     Why? 

37.  Find  the  time  between  4  and  5  o'clock  when  the  hands 
of  a  clock  are  directly  opposite  each  other. 

38.  John  could  remove  the  snow  from  a  walk  in  30  minutes. 
James  could  do  it  in  20  minutes.  John  began  the  work,  but 
later  James  took  his  place,  and  the  snow  was  all  removed  in 
25  minutes  from  the  beginning.     How  long  did  John  work  ? 

39.  A  could  dig  a  trench  in  15  days,  and  B  could  dig  it  in  2( 


i' 


LINEAR  EQUATIONS— ONE  UNKNOWN  NUMBER        177 

days.  If  they  worked  together,  how  long  would  be  required  to 
dig  it?  ^ 

«  40.  A  can  do  a  piece  of  woi^t  in  ten  days  ;  but  after  he  has 
worked  two  days,  B  comes  to  help  him,  and  together  they 
finish  it  in  three  days.  In  how  many  days  could  B  alone 
have  done  the  work  ? 

41.  A  solved  90%  of  the  problems  in  an  exercise,  and  B 
solved  I  as  many  as  A.  If  B  had  solved  6  more,  he  would 
have  solved  70%  of  all  the  problems  in  the  exercise.  How 
many  problems  in  the  exercise  ? 

42.  A  man  made  two  investments  amounting  together  to 
$6250.  On  the  first  he  gained  6%,  and  on  the  last  he  lost 
3%.  His  net  gain  was  $150.  What  was  the  amount  of  each 
investment  ? 

43.  If  12  lbs.  of  iron  weigh  11  lbs.  in  water,  and  20  lbs.  of  lead 
weigh  19  lbs.  in  water,  find  the  amounts  of  iron  and  lead  in 
a  mass  which  weighs  72  lbs.  in  air  and  68  lbs.  in  water. 

44.  In  an  alloy  of  gold  and  silver  Aveighing  60  oz.,  there  is 
5  oz.  of  gold.  How  much  silver  must  be  added  in  order  that 
10  oz.  of  the  new  alloy  shall  contain  but  \^  oz.  of  gold  ? 

•  45.  There  is  a  number  of  three  digits,  each  less  by  two  than 
the  one  to  its  right.  If  the  order  of  the  digits  is  reversed,  a 
new  number  is  obtained  whose  value  exceeds  that  of  the  given 
number  by  83  times  the  sum  of  its  digits.     Find  the  number. 

Note. — In  algebra  two  general  numbers  written  side  by  side,  as  ah,  in- 
dicate the  product  of  the  factors  a  and  b.  In  the  decimal  system  of 
writing  definite  numbers  two  numbets  so  written  do  not  represent  a 
product.  By  the  place  value  principle  of  the  decimal  notation  the  value 
represented  by  a  figure  not  only  depends  upon  its  shape  but  upon  the 
place  it  occupies.  Thus,  37  means  3tens  +  7ones,  or  3x10+7. 
If  tens'  digit  is  represented  by  x  and  ones'  digit  by  ?/?  th^  value  of 
the  number  which  they  represent  is  lOx+y, 


CHAPTER  XIII. 

LINEAR  EQUATIONS-MORE    THAN    ONE   UNKNOWN 
NUMBER.    SYSTEMS. 

121.  Indeterminate  equations.  An  equation  which  contains 
two  or  more  general  numbers,  or  unknowns,  will  be  satisfied 
by  an  indefinitely  great  number  of  sets  of  values  of  these 
general  numbers.  Such  an  equation  is  called  an  indeterminate 
equation. 

Consider  the  equation  x  +  y=10.  Solving  it  for  a?,  we  have 
x=10—y.  Now  if  different  values  are  assigned  to  y,  as  many 
corresponding  values  are  obtained  for  x.  Thus,  when  1/= 0,  a?= 10 ; 
when  y=l^  x=Q\  when  y=2,  x=S;  when  i/=  — 4,  x=14;  when 
2/=12,  x=—2;  etc.  Since  the  number  of  values  we  may  thus 
assign  to  y  is  indefinitely  great,  the  number  of  sets  of  values  of  x 
and  y  which  satisfy  the  equation  will  be  indefinitely  great. 

122.  Solutions.  In  an  equation  containing  two  or  more  un- 
known numbers,  i.  e.,  an  indeterminate  equation,  the  sets  of 
values  of  the  unknown  numbers  which  satisfy  the  equation 
are  called  solutions  of  the  equation. 

Thus,  one  solution  of  y—^x=2  isa?=l,  y=5;  because  for  these 
values  of  x  and  y  the  equation  becomes  5—3=2,  an  identity. 

123.  Common  solutions  of  two  linear  equations  with  two  un- 
knowns. Two  linear  equations  which  involve  the  same  two 
unknown  numbers  will,  in  general,  have  one,  and  only  one, 
solution  common — i.  e.,  one  set  of  values  of  the  unknown  num- 
bers which  will  satisfy  both  equations. 

178 


MORE  THAN  ONE  UNKNOWN  NUMBER.     SYSTEMS     179 

This  may  be  illustrated  by  the  following  example.  Consider 
the  equations  x  +  y=G  and  x—y=2.     Some  of  the  solutions  are  : 

for  0^  +  2/^6,    ^^g^,    ^^,^,   ^^4^,    ^^3^,   ^^g[,   ^^^ 
x=6  )      ic=  — 7  )     £c=     8  I 

fnr^    «-2    ^=     0  )    a?-     1  )    x=2  \    x=3  I    jr  =  4  |    ic=5 
it-6  )     ir=7  1     x=8  I 

It  is  seen  that  of  all  of  the  solutions  here  calculated  there  is  only 
one  common  to  both  equations,  a?=4,  y=2. 

That  this  principle  is  true  in  general  will  be  shown  in 
Example  3,  §  127. 

Two  or  more  linear  equations  with  two  unknowns  may  have 
(1)  all  of  their  solutions  co7nmon ;  (2)  just  one  soliftton  com- 
mon  ;  or  (3)  no  solution  common. 

Two  such  equations  having  all  solutions  common  are  called 
equivalent  equations.  One  may  be  derived  from  the  other  by 
the  use  of  axioms. 

Thus,  2x—y=3  and  4:X=Q  +  2y  have  all  solutions  common,  and 
hence  are  equivalent.  The  second  may  be  derived  from  the  first 
by  multiplying  both  members  of  the  first  by  2,  then  addmg  2y  to 
both  members  of  this  equation. 

Two  linear  equations  having  no  solution  common  are  called 
inconsistent  equations. 

Thus,  a? +  22/ =8  and3a?+6«/=5  are  inconsistent  equations.  No 
solution  of  either  equation  will  satisfy  the  other. 

Two  linear  equations  having  just  one  solution  common  are 
called  independent  equations.^ 

*  Two  or  more  equations  which  liave  common  solutions  are  some- 
times called  simultaneous  equations. 


i 


180  ALGEBRA 

124.  Systems  of  linear  equations  with  two  unknowns.  A  sys- 
tem of  linear  equations  with  two  unknowns  is  a  group  of  two  or 
more  equations  which  contain  these  unknowns. 

Thus,  Qx  +  y=5  and  x—5y=4:  constitute  a  system. 

A  solution  of  a  system,  such  as  defined  above,  consists  of  a 
sohition  common  to  all  of  the  equations  in  the  system. 

Thus,  a  solution  of  the  system  <        ,  ^     .  is  x—4:,  ?/=0. 

Since  a  system  of  two  independent  linear  equations  with 
two  unknown  numbers  has  one  solution,  such  a  system  is 
called  a  consistent  or  determinate  system. 

Three  or  more  linear  equations  which  contain  the  same  two 
unknown  numbers  have,  in  general,  no  common  solution.  A 
system  of  such  equations  is  called  an  impossible  system. 

i2x+y=l{),  (1) 
Thus,  in  the  system    <  3a? — y=  5,  (2)  the  only  solution  common 
(    x  +  y=  2,  (3) 

to  (1)  and  (2)  is  x=^^  y—^\  and  this  solution  will  not  satisfj^  (3). 

125.  Equivalent  systems.  Two  systems  of  equations  which 
have  the  same  solutions  are  called  equivalent  systems.  Since 
two  equivalent  equations  have  all  of  their  solutions  common, 
it  follows  that  two  systems  are  equivalent  if  the  equations  of 
one  system  are  equivalent  to  the  equations  of  the  other  system. 
In  general,  two  systems  of  linear  equations  will  be  equivalent  if 
the  equations  of  one  are  derived  from  the  equations  of  the  other. 

For  example,  the  only  solution  of  the  system    ■!  U  J^g^^s' 
is  a?=2,  2/ =3.     Adding  the  corresponding  members  of  (1)  and  (2), 
we  get  a  new  equation  (3)  3a? +32/= 15.     And  subtracting  the 
members  of  (2)  from  the  corresponding  members  of  (1),  we  get 
another  new  equation  (4)  x—y—  —  l. 

Equations  (3)  and  (4)  form  a  new  system  \  ^Z^,f^},\'  which  is 

(     X     y —      1, 


MORE  THAN  ONE  UNKNOWN  NUMBER.     SYSTEMS     18l 
equivalent  to  the  old  system,  because  its  only  solution  is  also 

It  follows  that  a  system  of  equations  may  be  solved  by 
solving  an  equivalent  system. 

126.  Elimination.  A  system  of  two  linear  equations  with 
two  unknowns  is  solved  by  a  process  called  elimination.  This 
process  consists  of  combining  the  two  equations  of  the  system 
so  as  to  obtain  a  new  equation  which  contains  but  one  unknown 
number. 

Tliere  are  three  principal  methods  of  elimination  :  (1)  by 
addi'ion  or  subtraction,  ('2)  by  comparison,  (3)  by  substitution. 

We  now  proceed  to  discuss  these  three  methods  of  elimina- 
tion in  solving  systems  of  linear  equations  with  two  unknowns. 

127.  Elimination  by  addition  or  subtraction. 

Example  1.     Solve  the  system    j  l^^^^^^l'  ^11 

Let  us  first  eliminate  y. 

Multiplying  (2)  by  4,  12x-{-4y  =  36.  (3) 

Subtracting  (1)  from  (3),  7x=U. 

Hence,  x=2. 

The  value  of  y  may  now  be  found  in  like  manner  by  eliminating 
X  between  equations  (1)  and  (2). 

Or,  replacing  x  by  its  value  2  in  equation  (1),  we  have 
10  +  42/=22. 

Hence,  2/=3. 

Let  the  student  see  if  the  solution  x=2,  y=3  satisfies  both 
equations  of  the  given  system. 

Notice  that  the  system  (1)  and  (2)  was  solved  by  solving  the 
system  (1)  and  (3).     See  §125. 

This  example  illustrates  elimination  by  subtraction. 

Example  2.     Solve  the  system        ]  4x  +  3w=  — 3  (2) 

We  may  first  eliminate  either  x  or  y.     Let  us  eliminate  y. 


182  ALGEBRA 

Multiplying  (1)  by  3,  21x-6?/=93.  (3) 

Multiplying  (2)  by  2,  8x  +  6y=-Q.  (4) 

Adding  (3)  and  (4),  29^=87. 

Hence,  •        x=^. 

The  value  of  y  may  now  be  found  by  eliminating  x  between 
equations  (1)  and  (2). 

Or,    replacing  x  by   its   value   3   in   equation  (2),   we  have 
12  +  3y=-S. 

Hence,  y=  —  5. 

The  system  (1)  and  (2)  was  solved  by  solving  the  equivalent 
system  (3)  and  (4). 

This  example  illustrates  elimination  by  addition. 

The  method  used  in  the  above  examples  would  apply  to  any 
system  of  linear  equations.  Hence  we  have  the  following  rule 
for  elimination  by  addition  or  subtraction : 

3fulti2:)ly  the  memhers  of  each  equation,  if  necessary,  by  such 
a  iiumher  as  loill  make  the  absolute  value  of  the  numerical  co- 
efficients of  one  of  the  unknown  numbers  the  same  in  both  of  the 
resulting  equations.  Add  or  subtract  the  corresponding  members 
of  the  resulting  equations. 

Note. — In  any  method  of  elimination  we  are  concerned  witli  only  the 
common  solutions  of  tlie  equations.  Hence,  in  the  addition  or  sub- 
traction, which  is  involved  in  tliis  first  method  of  elimination,  it  is  as- 
sumed that  the  unknown  numbers  have  the  same  values  in  each 
equation  of  the  S3^stem.  Eor  example,  x  stands  for  the  same  value  in 
each  equation.  Otherwise  the  addition  or  subtraction  would  not  be 
allowable.     The  same  facts  apply  to  every  method  of  elimination. 

Example  3.     Show  that  the  system  of  general  equations  in 

X  and  7/,    ]  ^^i^^ZS'    has,  in  general,  one  and  but  one  solution. 

Given                                \  ax  +  hy=c,  (1) 

•.    ^'""^^                               \dx  +  ey=f.  .(2) 

Multiplying  (1)  by  d^adx  +  bdy=cd.  (3) 

Multiplying  (2)  by  a,  adx  +  aey=af.  (4) 

Subtracting  (4)  from  (3),  bdy—aey=cd—af.  (5) 


MORE  THAN  ONE  UNKNOWN  NUMBER.    SYSTEMS     183 

Now  (5)  is  a  linear  equation  in  the  one  unknown  number  y. 
We  learned  in  Chapter  XII  that,  in  general,  such  an  equation 
has  one,  and  only  one,  solution. 

Solving  (5)  we  get  y=M=^e 

Similarly,  by  eliminating  ?/,  we  get 

aex—bdx=ce—bf.         (6) 

/^g Tf-f 

Solving  (6)  we  get  ^=^^3^^- 

Hence,  in  general,  the  system  has  one  and  but  one  solution  : 

^_ce-bf  cd-af 

ae—bd'  bd—ae  ^ 

Note. — It  will  be  found  that  if  certain  relations  exist  between  the 
general  co  efficients,  a,  b,  c,  d,  e,  and/,  this  conclusion  fails  ;  i.  e.,  the 
equations  are  equivalent  or  inconsistent. 

EXERCISE   57. 

Eliminate  by  addition  or  subtraction,  and  solve  the  following 
systems  of  equations  : 


l-2a3-f2/=6.  ''•  \'lxVy=\.  "■  l8«=5^-ll. 

(2£c  +  5?/=15,  ^  (4a-36  =  l,  «  (  8a=a;  +  34, 

(3a;-4y-ll.  ^'  (3a-4^  =  6.  ^'  (6«  +  8a;=53. 

10. 


j7y^^  =  42, 
(3y-^-8  =  0. 

..      (5a-2^-35  =  0, 
/^^'    t^>  +  4a-25  =  0. 

12    n«^+iy  =  30,  -,     (1.5a^-3.7.y  =  5.4, 


184 

ALGEBRA 

J.     (3^-2^  +  4-0, 

.Q     (3^  +  2y=4, 
^^-    t4y-3a;  +  l=0. 

16     f2a3  +  y==35, 
^^'    |5a;-3y  =  27. 

on     f  -4a;  +  3y  =  45, 
'    ^^-    \      2y  +  6^-   4. 

.^     (5y-5a^  =  15, 
^'-    |3£c4-5//-71. 

21.^ 

1  +  ^-35, 

^i^4      ^' 

18.   ^  ^     ^ 

4  +  6-^3- 

[|  +  .==45. 
22     n0a^  =  2  +  2y, 

Solve  for  x  and  i/  : 

24     (  aa;  +  ^>y  =  a^  +  ^^ 
*       Xbx  +  ai/  =  2ab. 

25. 

I  ax  +  ai/ =  a^ -{- b. 

128.  Elimination  by 

comparison. 

EXAMPLE].     Solve                  |t^:^=?^6: 
Let  us  first  eliminate  x. 

(1) 
(3) 

Solving  (1)  for  x, 

^-34-2/ 
^        4     • 

(3) 

Solving  (2)  for  x, 

a?=16-42/. 

(4)- 

Hence,   comparing  the  two  values  of  x  given  in  (3) 

and  (4), 

by  Axiom  7, 

3^;?/-16     42/. 

(5) 

Solving  (5)  for  y, 

•    2/-3. 

Replacing  y  in  (2)  by  2,       8  +  a?=16, 
whence,  x=8. 


(6) 


The  above  example  reveals  the  following  rule  for  elimina- 
tion by  comparison : 

jSolve  each  of  the  two  equations  for  the  value  of  one  of  the 
unknown  niimbers^in  terms  of  tJte  other ^  and  equate  the  resulting 
valim. 


MORE  THAN  ONE  UNKNOWN  NUMBER.     SYSTEMS.     185 

Solve  for  the  unknown  that  gives  the  simplest  expression 
from  each  of  the  equations. 


EXERCISE  58. 

Eliminate  l)y  comparison,  and  solve  the  following  systems  of 
equations. 

^'    \x-y  =  ^.  L    3  4 


3£c-y  =  l,  r3m-4/2=10, 

2^  +  5y^41.  13.   Y'\^'\^n  =  \l, 


1 

(^  +  2y-4, 
(22/-£c  +  12 

\^x  +  y  =  ll.  ^  (3.75.^  +  2.5y  =  10.25, 


22/-£c  +  12  =  0.  .-     (£c  +  2//-3  =  0, 


2a-^>  =  5, 


{ 


15. 


1 


I  7a  +  ^>  =  265. 


16. 


2^3  '' 


+  26-25.  l^,y 

3"^2 


jj     j  10.T  +  3«  =  174, 

°-    1  3a3+10«-125.  1^     (14a+66=0, 

,  .     Lo       1  1  6^-46  =  46. 

g     (5a  +  2y-l,  ^ 

(  13«  +  8y  =  ll.  Solve  for  x  and  y  : 

10.   J  \2x—my=n. 


L7^2  +  ^^^-  in    C«a^+y==^, 


20. 


C  ax  +  y 
{bx  +  y- 

<x-a'y  =  0, 
"i  a;  +  %-l. 


186  ALGEBRA 

129.  Elimination  by  substitution. 

Example  1.  Solve    \^^ZZ=%.  S 

Since  x  and  y  are  to  have  the  same  values  in  both  equations,  x 
may  be  replaced  in  equation  (2)  by  the  expression  for  its  value 
found  by  solving  equation  (1).  The  process  of  replacing  a  number 
in  any  expression  by  another  expression  which  represents  its 
value  is  called  substitution. 
Solving  (1)  for  ir,  x=2  +  Qy.  (3) 

Substituting  2  +  6i/   for  x,  in  (2) ,  3?/  -  8  (2  +  62/) = 29 .  (4) 

Solving  (4),  y=  —  l. 

Substituting  this  value  of  y  in  (1),  07  +  6=2. 

Hence,  0?=  — 4. 

From  the  above  example  we  have  the  following  rule  for 
eliminating  by  substitution  : 

Solce  one  of  the  equations  for  the  value  of  07ie  of  the  imknown 
numbers,  in  terms  of  the  other  one,  and  substitute  this  value  in 
place  of  that  mimber  in  the  other  equation. 


EXERCISE  59. 

Eliminate  by  substitution,  and  solve  the  following  systems 


of  equations : 

.      (a  +  Z>  =  30, 
^'    ]  3a-2^=25. 

5     (3.T  +  22/  =  26, 

„     (3a-7a;-40  =  0, 
'*•    |4a-3.T=9. 

g     (7a;-9y  =  13, 
^'    |5a;  +  2y  =  10. 

3     (4a.— 5y  =  26, 
^'    |3.T-6y-15. 

7    U^+iy=i3, 

-      (3m +  7;?.  =  16, 
*;   |2m  +  5w  =  13. 

g     (  7a^  +  4y  =  l, 
^'    t9^  +  4y  =  3. 

9. 

(3ic- 
(19a; 

11^  =  0, 
-19y  =  8. 

10. 


11. 


MORE  THA.N  ONE  UNKNOWN  NUMBER.     SYSTEMS     187 

^^-    t-4«c  +  6y=10. 

.«     (55a;-15y  =  270, 
^^'    (3l£c+19y  =  262. 


3  2 


.^  j22a;-5y  =  213, 

2a-7_13-.y  ^'*  |6£c-22/  =  51. 

j3     (4..  +  3y=22,  19.  i^  +  y^ff. 

.-      (5aj+ll2/=102,  „  (3^  +  2.y-42, 

^*-    \x-Si/  +  lQ  =  0.  '*"•  |13^  +  23y  =  225. 


EXERCISE  60. 

In  solving  the  following  systems  choose  the  method  of  elimi- 
nation that  seems  to  you  best  adapted  to  each  particular 
system. 

(x+i/  =  10,  ^     (8a;+6y=10, 

(ar-y  =  4.  **•    |5£c  +  2y  =  l. 

(8a;  +  i/  =  60,  n     (5a3-2y/-63, 

^-    |7a3-10y/:-9.  ^*    t2aj  +  y:-18. 

3     (7^+y  =  42,  10.    -[r^^^^n' 

.,  (i8.-20y^i,        11.  {1:^=11^ 

(1^/— 1^  =  2 

l5«-2y  =  14.  f7^-3y  =  26, 

„     (21ffl  +  86=-66,  '■''■   l2x-2y  =  ^. 

"•    t.49a-15S=-53. 


{ 


7x-4.i/  =  12,  14.    -^  4      2~^' 

Sx-bi/=0.  ( 2x-2i/  =  W. 


188 


ALGEBRA. 


16.  J 


-g-+y=18. 


Solve  for  x  and  y : 

^g     f«a5  +  6y  =  l, 
|&a?4-ay=l. 

jQ     S  hx-ay  =  h\ 
\ax—hy  =  a^. 

20.    i«f-%=«'-*', 
\^cry—ox=\j. 


16     j2y  +  79=5.x, 


17. 


22. 


2^      {2px  +  ^y  =  4:p'  +  q\ 
\x~'Zy  =  1p'-q. 


[^3"^4' 


6. 


130.  Systems  of  fractional  equations.  Certain  systems  of 
fractional  equations  may  be  solved  by  the  methods  of  this 
chapter.  Such  equations  should  not  be  cleared  of  fractions,  for 
the  resulting  equations  would  not  be  linear.  Also  clearing  of 
fractions  would  hitroduce  new  solutions,  ?*.e.,  would  give  solu- 
tions which  would  not  satisfy  the  original  system. 


ExAiMPLE  1.     Solve 


Multiplying  (1)  by  2, 


18_8 

a     h 

'2fi 
Subtracting  (3)  from  (2),       -^=6. 

Multiplying  by  5,  26  =  66, 

whence. 


=4. 


(1) 
(2) 
(3) 


81     36 


18. 


Multiplying  (1)  by  9, 


Multiplying  (2)  by  2,      f  +  f =30. 


(4) 

(5) 


Adding  (4)  and  (5), 


117 


=38, 


MORE  THAN  ONE  UNKNOWN  NUMBER.     SYSTEMS     189 


Ml] 

iltiplying  by  a, 

117= 38a, 

whence, 

117 
^-38- 

Example  2.    Solve 

3x  +  22/~^' 
^-?-3 

(1) 
(2)' 

9 
Multiplying  (1)  by  ^ 

3      27    45 

2x^Sy~  4' 

(3) 

Subtracting  (2)  from 

27      2      45 

Multiplying  by  242/, 
whence. 
Similarly, 

81  +  16  =  270?/- 

722/, 

EXERCISE  61. 

Solve : 

'•  i  i  J-i 

-«   y 

4. 

^ 

«  +  6-B'    ^ 
15     4_ 

7.  ^ 

r 1  ,  7  _5 

4^  +  %"8' 

1      3      5 

^'Ix    y'^28^ 

=  0. 

1  3^6     1 

6. 

J  •'■   y 

Ix    y 

8.   . 

57.  +  2Z."^' 

* 

3.  - 

^x     y 

6. 

- 

X      y             ' 
Lk     y 

9.  . 

fl-^  =4 

10.  < 

f-l     4-     ^--1 

l^x'V+y      ' 

Ll  +  ^    l+y    .2- 

11.  . 

^    4 

X- 
1 

3        9 

-1-8- 

131.  Systems  involving  three  or  more  unknowns.     It  is  easily 
shown  by  elimination,  as  in  Example  3,  §  127,  that  in  general 


190  ALGEBRA 

three  linear  equations  containing  the  same  three  unknowns,  or 
four  linear  equations  containing  the  same  four  unknowns,  etc., 
have  one  solution  common,  i.  6.,  one  set  of  values  for  all  of  the 
unknowns  which  will  satisfy  all  the  equations  at  once.  Hence, 
in  general,  a  system  of  linear  equations,  in  which  there  are  just 
as  many  unknowns  as  there  are  equations,  will  be  consistent  or 
determinate. 

If  there  are  more  unknowns  than  equations  in  the  system, 
in  general  the  number  of  solutions  of  the  system  is  indefinitely 
great.     Hence,  such  a  system  is  called  an  indeterminate  system. 

C      rjf  _i_  fj   t    2!  -—-  R 

Thus,  )  2x—y  +  z=^  ^^  indeterminate.  Some  of  its  solutions 
are  ^=1,  y=2^  z=3;  x=3,  y=S^  z=0;  a?=  — 1,  y=l,  z=6,  etc. 

And  if  there  are  more  equations  than  unknowns  in  the 
system,  in  general  no  solution  common  to  all  of  the  equations 
will  exist.     Hence,  the  system  is  an  impossible  system. 

To  solve  a  system  of  three  equations  containing  three  un- 
known numbers,  such  as  x,  y,  and  z,  we  may  (1)  eliminate  any 
one  of  the  numbers  from  any  two  of  the  equations ;  then  (2) 
eliminate  the  same  number  from  one  of  these  equations  and 
the  equation  not  used.  This  will  give  rise  to  two  new  equa- 
tions which  contain  only  two  unknown  numbers.  These  two 
equations  may  then  be  solved  as  a  new  system  by  the  methods 
of  the  preceding  sections. 

From  a  system  of  four  equations  with  four  unknown  numbers 
we  can,  in  like  manner,  derive  a  new  system  of  three  equations 
with  three  unknown  numbers ;  and  so  on. 

1  x  +  2y  +  2z=ll,  (1) 

Example  1.     Solve     ■<2x  +  y  +  z=7,  (2) 

{  3x  +  4y  +  z=U.  (3) 

By  subtraction  eliminate  x  between  (1)  and  (2). 

We  get  Sy  +  Sz=15.  ,    (4) 

By  subtraction  eliminate  x  between  (1)  and  (3). 


MORE  THAN  ONE  UNKNOWN  NUMBER.     SYSTEMS     191 

We  get  2y  +  5z=19.  (5) 

Now  by  subtraction  eliminate  y  between  (4)  and  (5). 

We  get  9^=27, 

whence,  z=3. 

Substituting  this  value  of  z  in  equation  (5), 
we  get  2i/  + 15=19, 

whence,  2/=^. 

Now  substituting  the  values  of  both  y  and  z  in  equation  (1), 
we  get  x  + 4  +  6  =  11, 

whence,  x=l. 


EXERCISE  62. 


^7 


Solve  the  following  systems : 

(x+y  +  z=9, 

1.  ■<  2ic+y— 2=0, 
(  3ic— y  +  s=5; 

(  x—2y  +  z=Q, 

2.  ■}x  +  Sy  +  2z  =  U, 
(2x-y  +  z=lS. 

3.  }  Sx  +  4y  +  6z^7, 

(ic  +  2y  +  62=4. 

(bx  +  6y  +  7z  =  S, 

4.  •}  10a;-i2y  +  2l2;=3, 
(  15.93— 6y  + 14^  =  4. 

(  Sx  +  y-z=2, 
6.    -\x-2y-Sz  =  Q, 
(y  +  z+l  =  0. 

(x  +  y  =  l, 

6.  ]y+z=9, 

(a;  +  s=— 6. 

C2x  +  y-z  =  7, 

7.  }y-x=l, 
(z-y  =  l. 


^  +  ^  +  ^=19 


8.   ^   ^_l-+^  =  6 
10     10^6     ^' 

L4  +  5     r5""^- 


9.  ^ 


10;«  +  -|  +  s  =  7, 

a;.+y-2s  =  16, 

X     y 


(2p-2^  +  3r  =  10, 
10.    \  ^}  +  q-r=b, 
'  (p-q  +  2r=7. 


11. 


12. 


x+y+z  +  w  =  10, 
X — y — z-\-w  =  0, 
2ic +  2/4-3^—10  =  9, 
Sx — y-\-z—2w=  —4. 

^2/9— </— r+s  =  13, 
j(>  +  5'— r— s=— 1, 
jL>— (Z  +  r— s=— 5, 

^Sp  +  2q~r  +  2s=-17. 


192  ALGEBRA 

132.  Geometric  picture  or  graph  of  the  equation.  We  have 
seen  that  an  indeterminate  equation  has  an  indefinitely  large 
number  of  solutions.  The  relation  between  these  solutions, 
which  is  expressed  by  the  equation,  may  be  more  vividly  re- 
presented by  means  of  the  graph  of  the  equation,  discussed  in 
the  following  sections. 

133.  Coordinates.  The  position  of  a  city  on  the  earth's 
surface  is  determined  by  its  longitude  and  latitude ;  i.  e.,  one 

can  locate  the  position  of  the 
Y  city,  if  its  longitude  and  lati- 

tude are  both  known.  Now  by 
the  method  of  §  33,  longitude 
east  may  be  called  negative 
,  longitude,  and  longitude  west 
may  be  called  positive  longi- 
tude. Also'  latitude  south  may 
be  called  negative  latitude,  and 
^  latitude   north   called  positive 

^^'    •  latitude.     Thus,  we  speak  of  a 

certain  city  as  being  —120°  longitude  and  +30°  latitude. 

In  like  manner,  the  exact  position  of  a  point  P  (see  Fig. 
1),  in  the  plane  of  this  j^age  may  be  determined,  if  we 
know  its  distances  and  directions  from  two  straight  lines  JCX 
and  YY\  drawn  at  right  angles  to  each  other  and  meeting  at 
O.  We  shall  represent  the  distance  from  P  to  the  line  YV 
i.  e.,  iVP,  by  x,  called  the  abscissa  of  the  point  P/  and  the 
distance  from  P  to  the  line  A'X^,  i.  e.,  MP,  we  shall  represent 
by  y,  called  the  ordinate  of  the  point  P. 

The  abscissa  x  and  ordinate  y  are  called  the  coordinates  of 
the  point,  and  the  lines  XJT  and  YY  are  called  the  axes  of 
coordinates.  A'X'  is  the  jr-axis  and  YY  is  the  /-axis.  The 
point  of  intersection  0  is  called  the  origin  of  coordinates. 


M 


MORE  THAN  ONE  UNKNOWN  NUMBER.     SYSTEMS     I93 


The  student  will  see  that  the  axes  XX'  and  YY'  correspond  to 
the  equator  and  the  prime  meridian  on  the  earth's  surface,  and 
that  the  coordinates  x  aiid  y  correspond  to  the  longitude  and 
latitude,  respectively,  of  a  place  on  the  earth's  surface. 

If  an  abscissa  drawn  to  the  rig/U  of  the  y-axis  (  YY')  is  called 
positive,  and  one  drawn  to  the  left  is  called  negative  ;  and  if  an 
ordinate  drawn  above  the  £c-axis  {XX')  is  called  positive,  and 
one  drawn  helow  it  is  called  negative  ;  then  the  exact  position 
of  the  point  is  known  when  its  coordinates  are  known. 

Thus,  to  locate  a  point  P  (See  Fig.  2)  whose  abscissa  is  + 1  and 
ordinate  +2,  measure  1  unit  to  the  right  of  O,  then  2  units  up 
from  XX' .  This  point  is  usually  re- 
presented by  the  symbol  (1,  2),  or 
P(l,  2).  Similarly  to  locate  the  point 
Q{—\,  1),  measure  \  unit  to  the  left  of 
O  and  1  unit  up  from  XX'.  To  locate 
P(— 1,  —2),  measure  1  unit  to  the  left 
of  O,  then  2  units  down  from  XX' . 
To  locate  >S(1,  —1),  measure  1  unit  to 
the  right  of  O,  then  1  unit  down 
from  XX'. 

Let  the  student  draw  a  figure  and 
locate  the  following  points  :  (—1,  2); 
(3,  -2);  (1,3);  (-2,  -3). 


Q(-M 


H 


Rf-A-^J 


y' 


a^j 


X' 


s(''-o 


Y 

Fig.  2. 


134.  The  graph.     The  graph,  or 
geometric  picture  of  an  equation,  is 
the  line  upon  which  are  situated  all  of  those  points  whose  co- 
ordinates, represented  by  x  and  y,  satisfy  the  equation. 

Consider  the  equation  x—y=2.  By  assigning  values  to  y  and 
computing  the  corresponding  values  of  x,  we  find  a  few  solutions 
as  follows : 

x=2  I     x=S  I     x=4t  I     x=^  \     x=Q  } 
y=0\     y=l\     y=2\     y  =  3\     y=4.\ 
x=     1)      x=     0)      x*=-l  )      x=—2) 
y=.-l\     2/=-2i     y=^-d\     V  =  -^V 
13 


194: 


ALGEBRA 


Locating  the  points  whose  coordinates  are  these  sohitions,  we 
get  a  series  of  points  as  in  Fig.  3.  It  is  seen  that  these  points  are 
not  scattered  at  random  over  the  page,  but  all  appear  to  lie  upon 
one  straight  line,  MN.  Hence,  the  straight  line  MNis  the  geo- 
metric picture  or  graph  of  the  equation  x—y=2. 


r 

N 

/ 

A 

^v^ 

J 

^.»/ 

^ 

\, 

(-X2) 

f^.^f 

\ 

/jj) 

X 

^ 

ii. 

/C"/ 

X' 

]^ 

^ 

/-n 

^ 

^ 

(^.-J/ 

M 

-0 

I 

/ 

M 

/ 

A 

V 

It  can  be  shown,  by  the  aid  of  geometry,  that  the  graph  of 
every  linear  equation  with  two  unknown  numbers  is  a  straight 
line.     We  shall  assume  this  principle  in  our  work. 

Since  a  straight  line  can  be  drawn  when  two  points  on  it  are 
known,  to  draw  the  graph  of  a  linear  equation,  we  need  to  find 
only  two  solutions,  and  locate  the  two  points  whose  coordinates 
are  these  solutions.  Then  draw  a  straight  line  through  these  two 
points.     Thus,  to  draw  the  graph  of  x  +  2y=l,  we  find  two  solu- 


MORE  THAN  ONE  UNKNOWN  DUMBER.     SYSTEMS     195 

tions  to  be  a?=4,  2/=— |,  and  x=  —  S,  y=2.  Locating  the  cor- 
responding points  (4,  —  I)  and  (—3,  2),  and  joining  them  by  a 
straight  line,  we  have  the  graph  PQ^  Fig.  3.  Any  other  point 
whose  coordinates  consist  of  a  solution  of  this  equation  will  be 
found  to  lie  upon  the  line  PQ. 

If,  upon  the  graph  P  §,  we  locate  the  point  A,  and  measure 
its  coordinates,  the  coordinates  Avill  be  found  to  be  x=  —  b, 
y='^.  These  coordinates  satisfy  the  equation  a? +  2y  =  l.  Like- 
wise, the  coordinates  of  a  second  point  B  of  the  graph  P  Q  are 
seen  to  be  93  =  6, 2/ =— 21.  These  coordinates  also  satisfy  the 
equation  x  +  '2i/  =  l.  Now  in  like  manner  it  will  be  found  that 
the  coordinates  of  any  point  whatever  of  the  line  P  Q  will 
satisfy  the  equation  x^-2y=l  of  which  P§  is  the  graph. 

It  thus  appears  that  not  only  do  all  of  the  points  whose  co- 
ordhiates  satisfy  a  linear  equation  lie  upon  a  straight  line^  called 
the  graph  of  the  equation  ;  hut  also  the  coordinates  of  all  points 
y^hich  lie  upon  the  graph  of  the  equation  satisfy  the  equation. 

Note.  This  principle  will  be  found  true  of  the  graph  of  an  equation 
of  any  degree.  Accordingly,  the  graph  of  an  equation  is  sometimes 
called  the  locus  of  all  points  whose  coordinates  satisfy  the  equation.  In 
analytical  geometry  this  principle  is  a  fundamental  notion. 

135.  The  graph  of  a  consistent  system  of  equations.     The 

solutions  of  each  of  the  two  equations  of  a  system  in  two 
unknown  numbers  are  represented  by  the  coordinates  of  the 
points  which  are  situated  upon  their  graphs.  Hence  their 
common  solution,  the  solution  of  the  system,  is  represented  by 
the  coordinates  of  a  point  which  is  on  both  graphs ;  i.  c,  the 
point  where  they  intersect. 

Consider  the  system       \  J^T^—a 
(  <ix+y — o. 


^  '--^y^     -^r^  ^ 


190 


ALGEBRA 


The  graphs  of  it'— y  =  2  and  2.^  +  y  =  6  are  the  Imes  MN'  and 
RS  (Fig.  4),  respectively.  And  the  solution  of  the  system, 
a;=2|,  y=\,  is  represented  by  the  coordinates  of  the  point  P 
where  these  lines  intersect. 


\ 

\j 

y' 

V 

"( 

\ 

\ 

N 

^ 

\ 

\ 

/ 

\ 

\ 

/ 

\ 

\ 

/ 

A, 

0 

A 

^ 

X 

57 

V 

/ 

\ 

\ 

/ 

\ 

\ 

/ 

\ 

\ 

/ 

\ 

\ 

m/ 

\ 

B 

\ 

s 

/ 

y 

\ 

K 

■    Fig.  4. 

It  follows  that  a  system  may  be  solved  by  drawing  accu- 
rately the  graphs  of  the  equations,  and  measuring  the  co- 
ordinates of  the  point  where  they  intersect.  Paper  for  this 
purpose  (called  coordinate  paper)  ruled  in  small  squares,  may 
be  obtained  from  a  stationer. 


MORE  THAN  ONE  UNKNOWN  NUMBER.     SYSTEMS     I97 


136.  Graphs  of  equivalent  and  inconsistent  equations.  Im- 
possible systems. 

Two  equivalent  equations  are  represented  by  the  same 
graph. 

For  example,  the  equivalent  equations  x—y=2  and  2x=2y  +  4:, 
by  actually  drawing  their  graphs,  will  be  found  to  be  represented 
by  the  same  line  MN,  Fig.  4.  This  conforms  to  the  definition  of 
equivalent  equations — equations  all  of  whose  solutions  are 
common. 


Two  inconsistent  equatio?is  are  represented  by  lines  which 
cannot  meet,  however  far  prolonged.  Such  lines  are  called 
parallel  lines. 


198  ALGEBRA 

For  example,  the  inconsistent  equations,  2a7+i/=6and4a?+2y=4 
are  represented  by  the  lines  RS  and  AB,  respectively,  in  Fig.  4. 
These  lines  cannot  meet,  however  far  prolonged,  which  is  the 
same  as  saying  that  the  equations  have  no  common  solution. 
See  §  123. 

Impossible  systems.  Since  an  impossible  system  in  two 
unknowns  consists  of  three  or  more  independent  equations, 
the  graphs  of  such  a  system  are  three  or  more  straight  lines, 
which,  in  general,  do  not  meet  in  a  common  point. 

For  example,  consider  tihe  system 

2x  +  y=4, 
x—2y=4:, 
3x+52/=15. 

The  graphs  of  these  three  equations  are,  respectively,  the  lines 
AB,  MN,  and  BS  of  Fig  5.  These  three  lines  meet  in  three 
distinct  points.     That  is,  they  have  no  common  solution. 

Other  cases  of  three  lines  would  be,  (1)  two  of  the  lines  parallel 
and  cut  by  the  third,  (2)  all  three  parallel,  (3)  all  three  intersect- 
ing in  the  same  point,  in  which  case  they  have  a  common  solu- 
tion. 

EXERCISE  63. 

1.  Draw  axes  and  locate  the  following  points  : 

(1,-3)  ;  (4,  2)  ;  (-3,  1)  ;  (-2,  -2)  ;  (2,  -3)  ;  (0,  2);  (3,  0); 
(0,  0). 

2.  By  assigning  values  to  x^  and  finding  the  corresponding 
values  of  y,  calculate  10  solutions  of  the  equation  3a;-f  2y  =  6. 
Locate  the  points  whose  coordinates  are  these  solutions  ;  then 
join  these  points.     What  kind  of  a  line  do  you  get  ? 

3.  Find  two  solutions  of  the  equation  2x—Sy  =  Q.  Locate 
the  points  whose  coordinates  are  these  solutions.  Draw  the 
straight  line  through  these  points.     Find  three  other   solu- 


MORE  THAN  ONE  UNKNOWN  NUMBER.     SYSTEMS     199 

tions ;  and  locate   the   corresponding  points.     What  do  you 
observe  about  these  points  ? 

4.  Draw  the  graph  of  4£e— 7y  =  28. 

5.  Draw  the  graph  of  5a;— 4?/ =  20, 

6.  Draw  the  graph  of  2a;  +  5y  =  10. 

\  ^Q     Draw  their  graphs.  Measure 

the  coordinates  of  the  point  where  these  graphs  cross.     What 
do  you  discover  ? 

Solve  the  following  systems  by  drawing  their  graphs; 
Check  your  solutions  by  solving  by  elimination. 

°'  I  ^x-y  =  12,         ^'   I  2x  +  Si/  =  4.         "'"•  |  2x  +  Si/  =  12. 

11.  Show  by  their  graphs  that  6a;— y/  =  10  and  5a;— y  =  2  are 
inconsistent. 

12.  Show  by  their  graphs  that  x-\^bi/—2  =  0  and  3a;=6  — 15y 
are  equivalent. 

13.  Show  by  their  graphs  that  3a;— y =6  and  a;  +  2y  =  4  are 
independent. 

14.  Show  by  their  graphs  the  geometric  meaning  of  the  fact 
that  the  equations  3a;  — 2y  =  6,  a;  +  y  =  7,  and  4a;— y  =  13  have  a 
common  solution. 

15.  Show  by  their  graphs  the  geometric  meaning  of  the  fact 
that  the  equations  3a;  — 2y  =  12,  4a;— 3y=  — 1,  and  2a;— y  =  ll 
have  no  common  solution,  i.  e.,  that  they  form  an  impossible 
system. 

16.  Interpret,  by  graphs,  the  system  a;  +  2y  =  8,  2a;  +  4y  =  10, 
anda;— 3y  =  6. 

17.  Interpret,  by  graphs,  the  system  a;— 2y=4,  2a;— 4y=5, 
and  4a;— 8y  =  7. 


200  ALGEBRA 

18.  Interpret,  by  graphs,  the  systeni  Sec— y  =  5,  x^5y=^7,  and 

137.  Problems  which  lead  to  systems  of  linear  eq[uations. 

A  problem  that  contains  just  as  many  independent  condi- 
tions as  there  are  unknown  numbers  may  be  solved  by  the  use 
of  systems  of  equations.  The  independent  conditions  give  rise 
to  as  many  independent  equations.  There  will  be  as  many 
equations  as  there  are  unknown  numbers.  Hence  they  will,  in 
general,  form  a  system  which  can  be  solved. 

Example  1.  Two  numbers  differ  by  4,  and  their  sum  is  10. 
What  are  the  numbers  ? 

Let  x=  the  greater  number,  and  ?/=  the  less. 

Then,  by  the  first  condition  of  the  problem,  we  have 

x-y=4t.  (1) 

And  by  the  second  condition,  we  have  x  +  y=10.  (2) 

Now  we  solve  (1)  and  (2)  as  a  system. 

Adding  (1)  and  (2),  .  2^=14, 

whence  x=7. 

Subtracting  (1)  from  (2),  2y=6, 

whence  2/= 3. 

'    Hence,  the  required  numbers  are  7  and  3. 

Example  2.  A  mmiber  of  two  digits  is  9  more  than  the  sum 
of  its  digits,  and  the  tens'  digit  is  less  by  2  than  the  ones'  digit. 
What  is  the  number  ? 

Let  x=  the  ones'  digit,  and  t/=  the  tens'  digit. 

Tlien  the  number  itself  will  be  represented  by  lOy  +  x.  (See 
note  at  end  of  Exercise  56.) 

From  the  first  condition  of  the  problem, 

10y  +  x=x  +  y-h9.        (1) 

From  the  second  condition  of  the  problem, 

y--=x-2.  (2) 

Transforming  these  equations, 
we  have  from  (1)  92/=9,  (3) 

and  from  (2)  y—x=—2.  (4) ' 


MORE  TITAN  ONE  UNKNOWN  NUMBER.     SYSTEMS      201 

From  (3)  y=l. 

Substituting  this  value  of  y  in  (4), 

whence  £C=3. 

Therefore,  the  number  is  13. 

Example  3.  The  value  of  a  fraction  equals  f .  If  its  numerator 
is  diminished  by  2  and  its  denominator  increased  by  5,  the  value 
of  the  resulting  fraction  will  be  ^.     What  is  the  fraction  ? 

Let  x=  the  numerator,  y=  the  denominator. 

Then,      '-=  fraction  required. 

y 

Hence,  from  the  first  condition, 

And  from  the  second  condition. 
Clearing  of  fractions,  (1)  and  (2)  become 

and 

Equations  (3)  and  (4)  form  a  system  whose  solution  is  ^^"=12, 
y=15. 

Hence,  the  fraction  is  jf. 

Example  4.  One  man  and  one  boy  can  do  a  piece  of  work  in 
3|  days  which  10  boys  and  6  men  can  do  in  i  day.  How  long 
would  it  take  one  boy  to  do  it  alone  ? 

Let  x=  number  of  days  it  would  take  one  boy  to  do  the 
work;  and  y=  number  of  days  it  w^ould  take  one  man  to  do  it. 

Then  in  one  day  one  boy   could   do    -   of  the  work,  and  oik 


X    4 
x-2     1 

y  +  5~2' 

(1) 

(2) 

5x=4y, 
x-y  =  9. 

(3) 

(4) 

man  could  do 

1 

y 

of  it. 

Hence, 

i  +  J=^,or,*3, 

(1) 

and 

15  +  «  =  1,    or  2. 

«     2/    i 

(2) 

Now   equations   (1)  and   (2)  form  a  system  whose  solution  is 
ifzzzio,  y=6. 


202  ALGEBRA 

Hence,  one  boy  could  do  the  work  in  10  days. 
Also  one  man  could  do  it  in  6  days. 


EXERCISE  64. 

By  the  use  of  systems  of  equations  solve  the  following : 
'    1.  Find  two  numbers  whose  sum  is  29  and  difference  13. 

2.  Find  two  numbers  whose  sum  is  9,  and  such  that  one  of 
them  exceeds  twice  the  other  by  27. 

*  3.  A  board  12  feet  long  is  to  be  cut  into  2  pieces  whose 
lengths  are  as  7  and  17.     How  long  are  the  pieces  ? 

4.  The  difference  between  two  numbers  is  72,  and  one  of 
them  is  4  times  the  other.     What  are  the  numbers  ? 
,    6.  Four  oranges  and  9  peaches  cost  25  cents,  and  9  oranges 
and  4  peaches  cost  40  cents.     What  is  the  cost  of  one  orange 
and  of  one  peach  ? 

,  6.  A's  age  exceeds  B's  age  by  15  years,  and  twice  A's  age 
exceeds  3  times  B's  age  by  9  years.     What  are  their  ages  ? 

7.  For  $372  I  buy  cows  and  hogs.  Each  cow  costs  $42,  and 
each  hog  costs  $8.50.  I  get  28  head  in  all.  How  many  of 
each  do  I  buy  ? 

*"  8.  I  change  $1.50  into  dimes  and  nickels.  There  are  25 
coins  in  all.  How  many  dimes  and  how  many  nickels  are 
there  ?  -^ 

9.  Five  years  ago  a  father  was  6  times  as  old  as  his  son,  and 
5  years  hence  the  father  will  lack  5  years  of  being  3  times  as 
old  as  his  son  will  be  at  that  time.     What  are  their  ages  ? 

*  10.  If  one  of  two  numbers  be  divided  by  12  and  the  other 
by  9,  the  quotients  will  be  equal ;  and  if  the  first  be  divided 
by  9  and  the  second  by  3,  the  sum  of  the  quotients  will  be  13. 
What  are  the  numbers  ? 


MORE  THAN  ONE  UNKNOWN  NUMBER.     SYSTEMS     203 

11.  Find  two  numbers  such  that  if  2  be  subtracted  from 
the  first,  and  4  added  to  the  second,  the  results  will  be  equal ; 
while  if  4  be  subtracted  from  the  first  and  7  from  the  second, 
the  first  remainder  will  be  twice  the  second  remainder. 

12.  There  are  two  numbers  whose  sum  is  34.  And  if  the 
greater  be  divided  by  the  less,  the  quotient  will  be  3  and  the 
remainder  2.     What  are  the  numbers  ? 

*  13.  Find  a  fraction  such  that  if  1  be  added  to  its  numerator, 
it  will  reduce  to  i,  and  if  1  be  added  to  its  denominator,  it  will 
reduce  to  i. 

14.  Find  the  fraction  whose  numerator  exceeds  i  of  the  de- 
nominator by  2,  and  such  that  if  the  numerator  be  subtracted 
from  the  denominator,  to  form  a  new  denominator,  the  fraction 
reduces  to  2i. 

16.  Find  two  numbers  such  that  the  sum  of  the  first  and 
i  of  the  second  is  38  ;  while  the  difference  between  the  second 
and  i  of  the  first  is  34. 

16.  Find  two  numbers  such  that  if  the  first  be  multiplied  by 
2  and  the  second  divided  by  2,  the  sum  of  the  results  will  be 
100 ;  and  such  that  twice  their  difference  exceeds  their  sum 
by  24. 

17.  A  certain  sum  of  money  was  divided  equally  among 
a  certain  number  of  people.  If  there  had  been  6  persons 
more,  the  amount  received  by  each  would  have  been  $6  less  ; 
and  if  there  had  been  2  persons  fewer,  the  amount  received  by 
each  would  have  been  14  more.  How  many  persons  were 
there,  and  what  did  each  receive  ? 

18.  A,  B,  C,  and  D  have  between  them  $1025.  B  has  twice 
as  much  as  A  ;  C  has  150  less  than  B  ;  and  D  has  as  much  as 
B  and  C  together.     How  much  has  each  ? 

19.  A  certain  number  of  two  digits  equals  6  times  the  sum 
of  its  digits ;  and  if  the  digits  were  interchanged,  the  resulting 


204  ALGEBRA 

number  would  be  less  than  the  given  number  by  9.  What  is 
the  number? 

20.  A  merchant  bought  12  carriages  and  8  sets  of  harness 
for  $2300.  lie  sold  the  carriages  at  a  gain  of  20%  and  the 
harness  at  a  gain  of  'l^^o-,  receiving  in  all  12770.  What  was 
the  cost  of  each  carriage  and  of  each  set  of  harness  ? 

»  21.  A  grocer  bought  lemons,  some  at  12  cents  a  dozen  and 
some  at  4  for  5  cents.  He  paid  for  all  $2.64.  He  sold  them 
at  20  cents  a  dozen,  clearing  $1.36.  How  many  lemons  did  he 
buy  at  each  price  ? 

22.  A  man  invested  $3650  in  three  sums.  The  first  earned 
him  4*^,  the  second  5%,  and  the  third  6%.  The  total  profit 
was  $195.  The  sum  which  yielded  6%  was  \  of  the  whole 
amount.     Find  each  of  the  three  sums. 

23.  A  company  of  men  rented  a  yacht.  When  they  paid 
their  rental  they  found  that  if  there  had  been  2  more  persons 
to  pay  the  same  bill,  each  would  have  paid  50  cents  less  than 
he  did  ;  and  if  there  had  been  2  fewer  persons,  each  would 
have  paid  $1  more  than  he  did.  Find  the  number  of  persons 
and  the  amount  that  each  paid. 

24.  A  sum  of  money  on  interest  amounted  to  $783.75  in  9 
months  and  to  $806.25  in  15  months.  Find  the  rate  and  the 
principal. 

Suggestion. — Let  x=  principal ;  y=  interest  for  1  year. 

25.  The  rainfall  in  a  certain  locality  one  year  was  i^  as 
much  as  it  was  the  year  after ;  and  the  total  rainfall  for  the 
two  years  was  18.5  inches.  Determine  the  amount  for  each 
year. 

26.  A  and  B  are  30  miles  apart.  If  they  travel  in  the  same 
direction,  A  overtakes  B  in  40  hours.  If  they  walk  toward 
each  other,  they  meet  in  Q^^  hours.     What  are  their  rates  ? 


MORE  THAN  ONE  UNKNOWN  NUMBER.     SYSTEMS     205 

^  27.  A  man  walked  19  miles,  part  of  the  distance  at  the  rate 
of  3  miles  an  hour,  and  the  remainder  at  the  rate  of  2  miles  an 
hour.  If  he  had  walked  2  miles  an  hour  when  he  walked  3, 
and  3  miles  an  hour  when  he  walked  2,  he  would  have  gone  21 
miles  in  the  same  time.  How  far  would  he  have  gone  if  he  had 
walked  3  miles  an  hour  the  whole  time  ? 
♦  28.  Two  trains  start  at  the  same  time  from  stations  500 
miles  apart,  and  meet  in  5  hours.  If  the  first  should  start  4| 
hours  earlier  than  the  second,  they  would  meet  in  three  hours 
from  the  time  that  the  second  train  starts.  What  is  the  speed 
of  each  train? 

•  29.  A  and  B  run  races  of  300  yards.  In  the  first  race  A 
gives  B  a  start  .of  60  yards,  and  wins  by  10  seconds.  In  the 
second  race  A  gives  B  a  start  of  15  seconds,  and  wins  by  40 
yards.     What  are  their  speeds? 

30.  A  train  traveled  a  certain  distance  at  a  uniform  rate.  If 
it  had  gone  12  miles  an  hour  faster,  the  journey  would  have 
required  2  hours  less ;  and  if  it  had  gone  4  miles  an  hour 
slower,  the  journey  would  have  required  1  hour  more.  How 
many  hours  does  the  journey  require  ? 

*  31.  A  man  who  can  row  6  miles  an  hour  down  stream  can 
row  2  miles  an  hour  up  stream.  What  is  the  speed  of  the 
current  ? 

32.  A  crew  rows  down  stream  6i  miles  in  an  hour,  and  re- 
turns in  4  hours  20  minutes.  What  is  the  speed  of  the 
current,  and  at  what  rate  could  the  crew  row  in  still  water  ? 

33.  Two  trains,  each  300  feet  long,  run  on  parallel  tracks. 
If  running  in  the  same  direction,  it  requires  20  seconds  for 
one  to  pass  the  other.  If  running  in  opposite  directions,  it  re- 
quires 4  seconds  for  them  to  pass.  What  are  the  velocities  of 
the  trains  ? 

34.  A  belt  runs  over  two  pulleys,  and  the  small    pulley 


206  ALGEBRA 

drives  the  larger  one.  The  small  pulley  makes  180  revolutions 
a  minute  more  than  the  larger  one.  If  the  large  pulley  were 
one-fourth  larger  than  it  is,  the  small  pulley  would  make  252 
revolutions  a  minute  more  than  the  large  one.  What  are  the 
velocities  of  the  two  pulleys  ? 

35.  A  and  B  together  can  do  a  piece  of  work  in  13i  days. 
After  they  have  worked  6  days  B  leaves,  and  A  finishes  the 
work  in  16i  days  more.  In  how  many  days  could  each  of  them 
do  it  alone? 

.  36.  A,  B,  and  C  together  can  do  a  piece  of  work  in  40  days. 
A  and  B  together  can  do  it  in  48  days.  B  and  C  can  together 
do  it  in  96  days.  Find  the  time  in  which  each  alone  could 
do  it. 

37.  A  man  and  a  boy  together  receive  137.50  wages.  The 
man  works  10  days,  and  the  boy  8  days.  The  man  earns  in  4 
days  13.50  more  than  the  boy  earns  in  6  days.  What  wages 
does  each  receive  ? 

38.  A  and  B  contribute  130,720  to  an  enterprise.  A  leaves 
his  money  in  the  business  10  months,  and  B  leaves  his  money 
in  the  business  2  years  6  months.  If  their  profits  are  equal, 
how  much  does  each  contribute  ? 

39.  A  farmer  has  enough  corn  to  last  his  hogs  for  a  certain 
number  of  days.  If  he  were  to  sell  5,  the  corn  would  last  10 
days  longer ;  but  if  he  were  to  buy  7,  the  corn  would  last  ten 
days  less.  How  many  hogs  has  he,  and  how  many  days  will 
the  corn  last  ? 

40.  If  the  altitude  of  a  rectangle  be  increased  4  inches,  and 
its  base  diminished  2  inches,  the  area  will  be  increased  22 
square  inches ;  and  if  the  altitude  be  increased  1  inch,  and  the 
base  diminished  1  inch,  the  area  will  be  increased  2  square 
inches.    Find  the  base  and  altitude  of  the  rectangle. 

41.  The  report  of  a  gunshot  travels  71.3  yards    with  the 


MORE  THAN  ONE  UNKNOWN  NUMBER.    SYSTEMS      207 

wind  toward  A  in  the  same  time  that  it  travels  68.7  yards 
against  the  wind  toward  B.  Four  seconds  after  the  discharge 
the  report  is  heard  at  A  and  at  B,  which  are  2800  yards  apart. 
What  is  the  velocity  of  the  report  in  still  air,  and  what  is  the 
velocity  of  the  wind  ? 

42.  A  tank  can  be  filled  by  two  pipes  in  8|  minutes.  If  the 
first  is  left  open  10  minutes  and  the  second  8  minutes,  it  will 
be  filled.     In  what  time  can  each  pipe  fill  the  tank  ? 

43.  A  merchant  has  two  kinds  of  wine.  If  he  mix  3  gallons 
of  the  poorer  with  7  gallons  of  the  better,  the  mixture  will  be 
worth  11.53  a  gallon ;  but  if  he  mix  7  gallons  of  the  poorer 
with  3  gallons  of  the  better,  the  mixture  will  be  worth  $1.37  a 
gallon.     What  is  the  price  of  each  kind  of  wine  ? 

44.  Two  vessels  contain  mixtures  of  water  and  wine.  In  one 
there  is  2  times  as  much  water  as  wine ;  in  the  other  there 
is  6  times  as  much  wine  as  water.  How  much  must  be  drawn 
off  from  each  to  fill  a  third  vessel  which  holds  10  gallons,  in 
order  that  its  contents  may  be  half  wine  and  half  water? 

45.  There  are  two  alloys  of  copper  and  silver,  of  which  one 
contains  3  times  as  much  copper  as  silver,  and  the  other  con- 
tains 5  times  as  much  silver  as  copper.  How  much  must  be 
taken  of  each  alloy  to  make  7  lbs.,  of  which  half  shall  be 
silver  and  the  other  half  copper  ? 

46.  Fifteen  pounds  of  tin  weigh  13  pounds  in  water,  and  15 
pounds  of  zinc  weigh  13.5  lbs.  in  water.  How  much  tin  and 
how  much  zinc  in  an  alloy  which  weighs  56  pounds  in  air  and 
49  pounds  in  water  ? 

47.  I  have  three  compounds  composed  of  different  metals. 
The  first  contains  7  parts  (in  weight)  silver,  3  parts  copper, 
and  6  parts  tin ;  the  second  contains  12  parts  silver,  3  parts 
copper,  and  1  part  tin;  the  third  contains  4  parts  silver,  7 
parts  copper,  and  5  parts  tin.     How  much  of  each  of  these  three 


208  ALGEBRA 

compounds  must  be  taken  in  order  to  form  a  fourth,  which  shall 
contain  8  ounces  of  silver,  3|  ounces  of  copper  and  4i  ounces 
of  tin  ? 

48.  A  certain  number  consists  of  three  digits  whose  sum  is 
9.  If  198  be  subtracted  from  the  number,  the  remainder  will 
consist  of  the  same  digits  in  a  reverse  order ;  and  if  the  num- 
ber be  divided  by  the  hundreds'  digit,  the  quotient  will  be  108. 
What  is  the  number  ? 

49.  Find  three  numbers  such  that  i  of  the  first,  i  of  the  sec- 
ond, and  I  of  the  third  shall  together  be  equal  to  62 ;  i  of  the  first 
i  of  the  second,  and  i  of  the  third  shall  be  equal  to  47 ;  and  i 
of  the  first,  i  of  the  second,  and  i  of  tlie  third  shall  be  equal 
to  38. 

50.  If  I  were  to  enlarge  my  field  l)y  making  it  5  rods  longer 
and  4  rods  wider,  its  area  would  be  increased  by  240  square 
rods ;  but  if  I  were  to  make  its  length  4  rods  less,  and  its  width 
5  rods  less,  its  area  would  be  diminished  by  210  square  rods. 
Find  the  present  lengtli,  width,  and  area. 

51.  Find  a  number  of  three  digits,  such  that  the  sum  of  tlie 
digits  shall  be  15 ;  the  sum  of  the  hundreds'  digit  and  the  ones' 
digit  shall  be  one  less  tlian  the  tens'  digit;  and  the  ones'  digit 
subtracted  from  four  times  the  hundreds' digit  shall  equal  the 
tens'  digit. 

EXERCISES   FOR   REVIEW   (IV). 

1.  What  is  meant  by  the  degree  of  an  equation  ?     Illustrate. 

2.  What  is  a  linear  equation  ? 

3.  Give  a  rule  for  solving  a  linear  equation..  To  what  ty2)e 
form  must  the  given  equation  always  be  reduced? 

4.  How  many  solutions  has  a  linear  equation  ?  What  other 
name  is  often  used  for  "  solution  ?  " 


MORE  THAN  ONE  UNKNOWN  NUMBER.     SYSTEMS     209 

6.  Solve  for  the  general  number : 

(a)  ^x-S(l-2x)  =  ^x. 

(b)  (a-S)(a-\b)^(a  +  2){a-l)-9. 
2b' -b  .      2b         ^b 


(^•) 


4b'-9'^2b-3     2b  +  d' 


6.  Solve  for  each  general  number  involved : 

(a)  a(a  +  x)—x{l  —  a)  =  a''-\-l. 

(b)  at—d=^a. 

(c)  i=prt\  d—vt. 


7. 

Solve : 

6^+1 

2t- 

-4 

2t- 

1 

15 

It- 

T6" 

5 

8. 

Solve  for  a : 

X  _b^ 

ab     ax 

=  x'- 

-b\ 

9.  Solve  for  x  : 

{x-a)(x-b)  +  (a  +  by  =  (x  +  a)(x  +  b). 

10.  ^=-  +  -1    solve  for  each  letter. 
f    P     ^ 

11.  1^"= J    solve  for  w,  g,  and  r. 

gr 

12.  {a)  C=^^.  (b)  C=j^^,  (c)  C=-^;  what  value 

n 
of  n  will  make  (a),  (6),  and  (c)  exactly  equal  ? 

13.  What  is  an  indeterminate  equation?     How  many  solu- 
tions has  it  ? 

14.  When  are  two  linear  equations  in  x  and  y  independent? 
When  are  they  equivalent?     When  are  they  inconsistent? 

14 


210  ALGEBRA 

16.  What  is  meant  by  a  root^  or  solution  of  a  system  of  two 
equations  in  two  unknown  numbers  ? 

16.  How  many  solutions  has  a  system  of  two  linear  equations 
in  X  and  y  ?  How  is  this  shown  by  the  graphs  of  the  equa- 
tions ?    Illustrate. 

17.  What  is  elimination  ?  How  many  methods  of  elimina- 
tion are  discussed  in  this  book  ?  What  are  they  called  ? 
Illustrate  each. 

18.  Solve  the  following  systems : 

(2a^-3y  =  18, 

19.  Solve  for  a  and  b;  also  for  x  and  y  : 

(  ax  +  y  =  b, 
\bx  +  y  =  a. 

20.  What  do  you  first  solve  for  in  the  following  ?    Solve 


21.  Solve  for  x  and  y  : 

ix  +  y  =  2a, 
\{a—b)x=(a  +  b)y. 

22.  Solve  for  x  and  y  : 

_^ y    _    1 

a-{-b     a—b     a-\-b^ 

«^    ,    .y  _   ^ 

\^a-\-b     a—b     a  —  b 

23.  What  is  the  graph  of  an  equation  in  two  unknowns  ? 


MORE  THAN  ONE  UNKNOWN  NUMBER.     SYSTEMS.     211 

24.  What  kind  of  equation  is  satisfied  by  the  coordinates  of 
all  of  the  points  on  a  straiglit  line  ? 

25.  Represent  graphically  the  solution  of  the  system 

(aj-3y  =  5. 

26.  What  does  the  graph  of  tlie  following  system  show  ? 

(  3a.— 2y  =  6, 
(6a;-5y  =  15. 

What  do  you  observe  of  the  relation  of  the  coefficients  of  the 
same  unknown  in  each  equation  ? 

27.  What  does  the  graph  of  the  following  system  show? 

(a^  +  3y  =  2, 
XSx  +  9ij  =  Q. 

What  relation  do  you  observe  between  the  two  equations  ? 

28.  What  does  the  graph  of  the  following  show  ? 

Do  you  observe  any  relation  between  the  two  equations  ? 

29.  Do  you  observe  any  relations  in  Exercises  26,  27,  and  28 
by  which  you  can  tell  the  class  of  the  system,  i.e.,  whether 
the  equations  of  the  system  are  identical,  equivalent,  or  in- 
dependent ? 


CHAPTER  XIV. 

SURDS  AND  IMAGINARY  NUMBERS. 

138.  Surds.  The  definitions  of  a  surd  and  of  an  irrationrJ 
expression  Avere  given  in  §  69,  which  the  student  should  now 
reviews  Any  rational  expression  may  be  written  in  the  form 
of  a  surd. 


Thus,  2=y4=|/8=y'64;  a  +  b=^/a'  +  2ab  +  b^  = 


Va^  +  Sa'b  +  Sab'  +  bK 

In  the  expression  al/^y  the  factor  |/^is  called  the  surd  fac- 
tor, and  a  is  called  the  coefficient  of  the  surd. 

An  expression  like  j/ic,  in  which  the  coefficient  of  the  surd 
is  unity,  is  called  an  entire  surd. 

139.  Surds  may  differ  in  order.  The  order  of  a  surd  is 
indicated  by  the  index  of  the  root. 

A  surd  of  the  second  order,  or  a  quadratic  surd,  is  one  whose 
index  is  2. 

A  surd  of  the  third  order,  or  a  cu"bic  surd,  is  one  whose  index  is  3. 

A  surd  of  the  fourth  order,  or  a  biquadratic  surd,  is  one 
whose  index  is  4 ;  and  so  on. 

A  surd  of  the  nth  order  is  one  whose  index  is  n. 

Thus,  1/2,  Va—b,  l/x*,  are  quadratic  surds; 


1/10,  l/a?^  — 2/^  V  x^y^,  are  cubic  surds: 


y  9,  i7 a6^  Va*  +  b*,  are  surds  of  the  fourth  order;  and 


ya,  V  a^—b^  yab^-^,  are  surds  of  the  nth.  order. 

212 


SURDS  AND  IMAGINARY  NUMBERS  213 

140.  A  surd  is  in  the  simplest  form  when  the  expression 
under  the  radical  sign  involves  no  fraction,  and  contains  no 
factor  the  indicated  root  of  which  can  be  extracted. 

Thus,  the  simplest  form  of  v  27  is  3v  3;  the  simplest  form  of 

y  x^y^z^  is  xzv  X]f-\  the  simplest  form  of  1/    -p   is   -V  d}h. 

141.  Reduction  of  surds  to  simplest  form.  Surds  that  are  not 
in  the  simplest  form  may  be  reduced  to  the  simplest  form  by 
the  principles  of  §  67  and  §  68.     There  are  two  cases : 

{a).  When  the  expression  under  the  radical  sign  is  free  of 
fractions,  use  the  principle  established  in  §  67  ;  that  is, 

n/— ;  M  —    n,-j- 

\ab  =  Va\/h. 

From  this  we  have  the  following  rule : 

Separate  the  exjpression  under  the  radical  sign  into  two  fac- 
tors^ one  of  vnhich  is  a  perfect  ponder  the  indicated  root  of  which 
can  he  extracted. 

Example  1.     Reduce  to  the  simplest  form  |    — 81. 

3  —  3, 


Factoring,  V  -81  =  1/  -27-3=1    -27^  3=-3l/3. 
Example  2.     Reduce  to  the  simplest  form  j /32a*6^c^. 


V  32a*6«c«=i/ 16a*6«c*-2c2=  l/'16a*6Vi/  2c2=2a62ci7  2c^ 


Example  3.     Reduce  to  the  simplest  form  V x^—^x^y^-xy^ 


l/x^— 2x'^y  +  xy'^  =  V  x{x^ — 2xy  +  y-)  =  \/x^ — 2xy  -\-y^\/x 
=  {x-y)Vx. 

(b).  When  the  expression  under  the  radical  sign  is  a  frac- 
tion, (1)  tnidtiply  both  numerator  and  denominator  of  the  frac- 
tion by  such  an  expression  as  will  make  the  resulting  denominator 
a  perfect  pouter  the  indicated  root  of  which  can  be  extracted. 


214  ALGEBRA 

(2)    Then^  proceed  as  in  case  («),  making  use  of  the  principle 
established  in  §  68,  i.  6., 

^    b     yb 

Example  1.     Eeduce  to  the  simplest  form  i/|. 

Example  2.     Reduce  to  the  simplest  form  —  i/  ^~,. 


^x    Vifz  _Sx  ySxy^z _3x   y  y''  ^ 

y  V   27a^-y  V  81^*    ~y  V  ilx'^^^^ 

Sx-,^/~i?    i,- Sx  y      ,-z 

Example  3.     Reduce  to  the  simplest  form  (a—h)  1/  -— - 


EXERCISE  65. 

Reduce  to  the  simplest  form  : 

1-  1/8:  11.  _3f  =135.  21.  i/|. 

2.  V  625:  12.  i/27^y:  22.  fi/V-. 

3.  pzzu.  13.  V4MF.  23.  ^y% 

4.  f  729:  '     14.  ^TQ^Fxy.  24.  i/^ 
6.  |/72.                      15.  y'-Q2bxy\  *' 

6.  i/TOS:  •     16.  1^16^^  25.  |/^^. 

7.  v500:  17.  y'^3a^«-^y".  ^ 

8.  3^/3927  18.  ^l28a^^-6-^^  ^^'  l/^' 

9.  5iM080:  19.  ^>2^  3^  s /^y^ 
10.  1*16327                 20.  i4.  2//]/  1?~- 


SURDS  AND  IMAGINARY  NUMBERS  215 


28-Vi7^. 

2y.  |/(«_ic)^ 

30. 

31.  iA~-'/. 

|/  4«£c^^  4  Vlax'y  +  12cea;y=^  +  4ay\ 

32.  2i/    1  ,. 

'«•  (/j+i- 

35 


■Vl-l^^. 


142.  Addition  and  subtraction  of  surds.  Only  surds  which, 
when  reduced  to  the  simplest  form,  have  the  same  surd  factor 
can  be  added  or  subtracted.  Otherwise  the  addition  or  sub- 
traction can  only  be  indicated.  Just  as  3<:<  +  2a==5«,  so 
Z\/x-\-^^/x-~b\/x. 

Hence,  surds  vnth  a  common  surd  factor  are  added  or  sub- 
tracted by  adding  or  subtracting  the  coefficients  of  the  surd  fac- 
tors and  affixing  to  the  residt  the  common  surd  factor. 

Example  1.     Find  the  value  of  i/32  + 1/50. 

1/32  + 1/50=41/2  + 5.,l/2=(4  + 5)  l/2=9l/2. 


Example  2.     Simplify   Va'bc-^Vab''c-\-Vahc'. 

Va?bc — vab^c  +  V  abc'^ = a^  V  ahc — V  V  ahc  +  c^  yabc 
=  (a''-b'-}-c')yabG. 

EXERCISE  66. 

Simplify  the  following : 

1.  1/T8-V8  +  1/32:  5.  2v  3  +  1/81-V3. 

2.  1/75-1/3-1/12:  6.  i/40-i>320-2i/5. 

3.  i/20  +  1/45-i/M:  7.  4i>}-y}. 

4.  1/6 -v"294  + 1/486 +  1/24:     8.  2i/^  +  3i/'^-l/^ 


7;s 


216  ALGEBRA 

9.  yT^^-2i/2^'  +  i/W. 


10.  i/81a'-4i/192a^  +  3i;648al 

12.  2|/V^+3v/aFc— i/VFc. 

13.  i/^-i/^*  +  V^.  17.  l/F+^l/7S-2l/}- 

14.  1   T  +  3^|_^3.  18.  I'S-V  i+vV?. 

15.  i/27i-i/8S+l/64^^         19.  i /20  +  3i/4-2r4. 

16.  3|/48-y^r_2|/|.  20.  vM  +  i/f+i/S". 

143.  Change  of  order.  A  surd  of  a  given  order  may  be 
changed  to  an  equivalent  surd  of  a  different  order  by  the 
principle, 

]'  0""=  \   a". 

That  is,  the  value  of  a  surd  is  not  changed  by  multiphjing  or 
dimdmg  the  index  of  the  root  and  tlie  exponent  of  the  poicer  by 
the  same  number. 

This  principle  may  be  established  as  follows : 

Let 

Then 
Hence, 
or 

Hence, 

Therefore, 

Example  1.     Express  Va^  as  a  sixth  root. 
Multiplying  index  and  exponent  by  2,  we  get 

\/a^=y'a^\ 


mx 

y  a"'  =  r. 

a"^=r"^^ 

§64. 

y  a''^=yr'"-% 

Axiom  6. 

a"=r'"- 

l/a«=l/r"', 

Axiom  6. 

=  r. 

7/6^=1/ «^. 

Axiom  7. 

Example  2.     Reduce  yaWc  to  a  tenth  root. 


SURDS  AND  IMAGINARY  NUMBERS  217 

Multiplying  index  and  exponent  by  5,  we  have 


ya^b*c=  y  {d'h'cf=Va'Wc\ 


ElXAMPLE  3.     Reduce  yx^y^z^  to  the  lowest  order. 
Dividing  index  and  exponent  by  2,  their  greatest  common  divi- 
sor, gives 

y^x'y^^'=y^3(^y^z. 

144.  Reduction  of  surds  to  the  same  order.  Surds  of  different 
orders  may  be  reduced  to  equivalent  surds  of  the  same  order 
by  the  principle  of  §  143.  Manifestly  the  simplest  common 
order  to  which  surds  may  be  reduced  is  the  least  common 
multiple  of  their  orders. 

Example  1.     Reduce  Va^b,  v'a■'&^  Va^U"  to  the  same  order. 

The  least  common  multiple  of  the  indices  is  12.  Reducing 
each  surd  to  an  equivalent  surd  of  the  twelfth  order, 


ya'b=Va^¥;  Vd'b^=ya''b^-  ya'b'=Va^b'\ 

The  values  of  surds  may  be  compared  by  first  reducing 
them  to  equivalent  surds  of  the  same  order,  and  comparing  the 
resulting  numbers  under  the  radical  signs. 

Example  2.     Compare  the  values  of  i/l5  and  y  Qo. 


1/15  =  1/  W=y  3375 ;   V  60=  ;/  60^= 1/  3600. 
Since  3600  is  greater  than  3375,  y  60  is  greater  thani/15. 

145.  Multiplication  of  surds.  The  product  of  surds  which 
are  of  the  same  order  may  be  found  by  the  principle  established 
in  §  67;  that  is, 

y^aiyj^yab'. 


Surds  of  different  orders  whose  product  is  to  be  found  must 
first  be  reduced  to  surds  of  the  same  order. 


218  ALGEBRA 


Example  1.     Find  the  product  of  y'2a^,  i   3a^  and  F  4a. 


V2a^=l   8a«;  ySa'=V9a*;  V  4a=V  16a^ 


Hence,  V2d'V 'Sa^ ■V4a=\/8a'' y  9a* Vl^a^ 


l/Sa:>-9a*lQa' 


=  V1152a'^ 


=2aU/lSa\ 

Example  2.     Multiply  21^5  + 5]/2  by  3V5-i\/2. 

2VE+5V2 
3|/5-4]/2 
6-5  +  15l/l0 

-  8t/10-20-2 


30+   71/10-40,  or  7l/l0-10. 

3  - 

Example  3.     Express  as  an  entire  surd  3y  4. 

The  coefficient  may  be  expressed  in  the  form  of  a  surd  of  the 
same  order  as  the  surd  factor. 

We  have  3 = V 3-^ = 1/27. 

Hence,  31^4=  \^21^/~4=  1/^27x4=  i/^I08. 

EXERCISE  67. 

Express  as  surds  of  the  same  order  : 

1-   l/a,  \^a\  2.  yx\  yx,  V'x\       3.  i/5,  1^2,  1^3. 

4.  ]>7,  //I2;  1/2.  5.  l/^\  1^^,  l/V^ 

Which  is  the  greater : 
6.  1/5  or  v^lT?  7.  V  2210,  l'/320,  or  1^48? 


SURDS  AND  IMAGINARY  NUMBERS  219 

Express  as  entire  surds : 

8.  21^5.  10-  V7.  12.  i^^. 

9.  8i/8.  11.  ||/3.  13.  «T/X 


3^  "•  **'.  ^y  YT- 

27a;' 


^^-  Fl/«-  1^'  2^V  27^ 


Simplify : 

16.  i/2v^3v  4.  18.  i/61/'4l>'2.  20.  i/^T^by/^Tl. 

17.  2l^2-3l/5.  19.  1/^51/101/5.        21.  i  a^i/^^^l 

22.  i?^«^  +  2a^>  +  ^2v'^Hl>.  23.  (^/2+i/3)(^2-i/3). 

24.  (l  +  i/2-i/3)(l  +  i/2+i/3). 

25.  (|/e  +  i/2)(i/6-i/2). 

26.  (^3  +  i^2)(i>'9-i^6  +  f'4). 

27.  (1  +  1/2  +  1/3)^ 

Simplify  by  reducing  to  lower  order : 

28.  y'lM'.  31.  jyg^^v".  33     VWy 

29.  ^2W.  ./-.  'V^^^ 

_____  32    i/^  34    ,«/l^^«"^'' 

30.  i/Sia^'^^v^  *  K  F  '  V  ^uy^' 

35.  i/(^.         36.  ^s/z?:. 

146.  Two  binomials  which  contain  surds  of  the  second 
order  and  which  differ  only  in  the  signs  of  the  surd  terms  are 
called  conjugate  surds. 

Thus  1/2  +  1/5  and  i/2— 1/5  are  conjugate  expressions. 

Since  conjugate  surd  expressions  are  of  the  form  a  +  6  and 
a— 6,  the  product  of  a  pair  of  conjugate  expressions  is  a  rational 
expression.     Thus,  (a  +  y'h)  x  (a— i  /&} =«^— 6, 


220  ALGEBRA 

By  grouping  terms,  polynomials  involving  surds  of  the  second 
order  may  often  be  expressed  as  conjugate  expressions. 

Thus,  l/2  +  V^S+V5  and  i/2-i/3+i/5  may  be  written 
(l/2+ 1/5)  +  VS  and  (V2  +  |/5)-i/3. 

147.  Division  of  surds.  The  quotient  of  one  surd  by  an- 
other surd  of  the  same  order  may  be  found  by  using  the  prin- 
ciple established  in  §  68 ;  that  i ;, 

1/6      ^    b 

But  the  quotient,  where  the  divisor  is  a  surd  expression,  is 
reduced  to.  its  simplest  form  by  first  multiplying  the  dividend 
and  divisor  hy  such  an  exjyression  as  will  make  the  divisor 
rational. 

This  process  is  called  rationalizing  the  divisor.  The  mul- 
tiplier used  is  called  the  rationalizing  factor. 

Example  1.     Divide  61   2  by  v  3. 

Multiplying  both  dividend  and  divisor  by  \   3,  we  have 
6v /2_6i/2i/3^6| /6_3  ^g 
1/3       1    3F3         3 

Whe7i  the  divisor  is  a  binomial  involving  surds  of  the  second 
order  ^  the  simplest  rationalizing  factor  is  its  conjugate.  Why? 
(See  §  146.) 

Example  2.  Simplify    1^g+l^^_. 
2i/3-|-3i/2 

The  conjugate  of  2i/3  +  3l/2  is  2|/3-3l/2. 
Multiplying  by  this,  we  have 

l/2  +  i/3^_  (i/2-n/3)(2i,/3-3i/2)_  __:zi^__  1  ^/e 
2i/3-H3l/2      (2l/3-J-3l/2)(2i/3-3l/2)  ~    -6  "^ 


SURDS  AND  IMAGINARY  NUMBERS  221 

When  the  divisor  is  a  polynomial  which  involves  surds  of  the 
second  order,  group  terms  so  as  to  form  a  binomial,  and  then 
multiply  by  its  conjugate.  This  result  may,  likewise,  be  ex- 
pressed as  a  binomial  and  multipliedby  its  conjugate;  and  so  on 
until  the  divisor  is  rationalized. 

Example  3.     Divide  2  by  1/5  +  t/2— 1/3. 
2 _2 

V5  +  V2-  V  3  ~ (1/ 5  +  V  2)  - 1/3 

2(1/5  +  1/2  +  1/3")  


(1/5  + 1/2-1/ 3)(l/5  + 1/2  +  1/3) 

_2(l/5  +  i/2j-y3)     ^j,  l/5  +  l/2V|/3 
4  +  21/10         '  Z+VlO 

_(l/5  +  1/2  +  l/3)(2-  l/lO) 
(2  +  l/l0)(2-l/10) 

_  2 1/3 -3 1/2 -1/30 
-6 

EXERCISE  68. 

Simplify : 

1  i  6  ^  11     l/"T+^^   +^ 
'  1/2*                            •  1/-2  +  1/3'  y(l+x')-x 

2  V^  1/3-1/5  1/^+6  + i/^F^ 
*  1/3*                             '  1/3-1/2*                     i/^T7>-v/a=^ 

_    l/'2-i/3  „    2|/7  +  3i/2  ,„    i/^Ta-i/^ 

«•    ■ ::: •  O.      =: —-  !«• 


1/2  +  1/3 

1/3-1/5 

1/3-1/2 

2|/7  +  3i/2 

3v/2  +  2i/3 

3i/5-5i/3 

3|/5  +  5i/3 

a'' 

1/7  _  ■  •  3v/2  +  2i/3  |/ic  +  3  +  i/a; 

4    A.  o    3v/5-5i/3  14.         ^t^^^     . 

T/2  ^'  3|/5  +  5v/r  1  +  1/2  +  1/3 

^      12  a'  ,.  1/5-1/3 

5.  — ^'  10 .  10.  — = =: — • 

v3  *+l/(6^-a^)  v/5-1/3-1 


222  ALGEBRA 

16. 

17. 


y'2  +  yS  +  y7 

T/2 


18. 
19. 

yx+y 
Va 

y- 

-yz 
h 

l/a+|/ 

b- 

-yc 

y2  +  y3  +  yb 

148.  Powers  of  surds.  Powers  of  surds  are  obtained  by  the 
use  of  the  following  principle  : 

(,'/iy=l'/V. 
To  establish  this  principle,  we  have 

{l^ciy=\ya-{/a-i/a to  r  factors 

=  Paaa to  r  factors     §  67. 

3> 

Example  1.     Raise  2ay2a^b^  to  the  fourth  power. 


(2aV2a'b'Y=16a*y  {2a'by=16a'y  IQa'b"^, 

149.  Roots  of  surds.  Roots  of  surds  maybe  found  by  the 
principle, 

y{\/a)=ya. 

This  law  follows  from  the  fact  that  to  take  the  mth  root  of 
the  nth  root  of  a  is  to  take  one  of  the  m  equal  factors  of  one  of 
the  71  equal  factors  of  a,  which  is  one  of  the  mn  equal  factors 
of  a. 


Example  1.    Find  the  fifth  root  of  y^a^if. 

Example  2.    Simplify  v^{2a''bySa¥), 
First  express  2a^b^/3ab^  as  an  entire  surd. 

y'{2a'by'3a^==  V^{l/T2a^')  =  y'na'b^ 


Simplify  : 

1.  (1/2)1 


SURDS  AND  IMAGINARY  NUMBERS 
EXERCISE  69. 


^23 


2.  iY'lx'yy. 

3.  (5a;'^i/2^')l 

4.  {-'lah^/^hy. 

6.  (^Wby. 


7.  {Aayya^y. 

8.  (3«yv^^*)l 


12.  i7(^/T6). 

13.  i/'(|^625<). 


9.  {4:\/d'-by. 

10.  i^(i/5). 

11.  iV{i>(^2)}. 


14.  y\i/21xY)' 

15.  \/{VM). 

16.  1/^(8^4^). 


17.  1/  /(a^-^>'^)^     18.  ^  ;>8^. 


19.  i/  49icV«'|?'8a^ 


20.   {^{^(f  «"')}• 

150.  Imaginary  numbers.*  In  §  65,  even  roots  of  negative 
numbers  were  called  imaginary  nmnhers^  and  it  was  shown 
that  these  numbers  did  not  belong  to  the  series  of  numbers 
with  iMch  ice  were  then  acquainted.  They  constitute  a  new 
series  of  numbers  with  some  pecidiar  properties. 

By  means  of  more  advanced  mathematics  any  imaginary  num- 
ber may  be  expressed  in  terms  of  quadratic  imaginary  numbers, 
i.  e.,  square  roots  of  negative  numbers.  Hence,  in  this  chapter  all 
of  the  language  shall  have  reference  only  to  quadratic  imaginary 
numbers. 

For  the  sake  of  distinction,  all  numbers  heretofore  considered 
are  called  real  numbers.  Numbers  of  the  form  i  —  ^,  where  b 
is  positive,  are  sometimes  called  pure  imaginary  numbers. 
Expressions  of  the  form  a+|/^^  when  a  is  real  and  y^^b 
a  pure  imaginary  number,  are  called  complex  expressions. 

*  We  have  seen  that  the  first  numbers  known  were  whole 
numbers ;  that  by  extending  the  process  of  division  fractions  were 
conceived  ;  that  by  extending  the  process  of  subtraction  negative 
numbers  were  conceived  ;  and  tliat  by  extending  the  process  of  evo- 
lution surds  and  finally  imaginary  numbers  were  conceived.  In  the 
investigations  of  higher  mathematics  the  imaginary  number  plays  an 
important  part. 


224  ALGEBRA 

151.  If  a=0,  the  complex  expression  a+j/^—b  is  a  pure 
imaginary  number  i^  —  b;  but  if  b  =  0  instead,  the  expression 
becomes  a  real  number,  a.  It  thus  appears  that  real  numbers 
and  imaginary  numbers  are  both  particular  kinds  of  complex 
expressions. 

The  squares  of  all  real  numbers  are  real  positive  numbers^ 
and  the  squares  of  all  pure  imaginary  n^imbers  are  real  negative 
numbers. 

Thus,  (  +  6)2= +  3G;  (-4)2= +  16;   (  +  l/2)2=+2;    (-l/2)2=  +  1^4. 
And  (l/"^2^-3;   (-|/^)2=-5;   (-4l/^)2=-32. 

152.  Typical  form  of  imaginary  numbers.  By  the  principle 
]/«3=]«]/ 5  any  imaginary  number  can  be  expressed  in  the 
form  c]/^^,  called  the  typical  form. 


Thus,  i/-4=i/4(-l)  =  i/4i/-l=2|/-l. 

And  -|/-10=-i/10-(-1)  =  -i/10t/'^1. 

Hence,  the  properties  of  any  imaginary  number  may  be 
studied  by  studying  the  properties  of  v^^^,  usually  denoted 
by  *,  which  is  taken  as  the  imaginary  unit. 

Note. — The  principle  \/ah—\/a\/b  does  not  hold  when  \/ a  and  \/b 
are  both  imaginary,  i.  e.,  when  a  and  h  are  both  negative.  Thus, 
l/^}/— 9   does   not   equal  i/36,  or  6,  but  =i/a6(i/=4)2,  or  -6. 


153. 

Powers  of  ] 

-1 

or  /.     I 

^or 

the  s 

we  have 

i- 

=  y  - 

"1; 

By  § 

64,  r'  = 

-{V- 

—  1  \2  = 

=  -1; 

i^-- 

=  {V~ 

— ~1V  = 

=i-i'=- 

-*. 

i'- 

=  {V~ 

—  1V  = 

=i-e=- 

~f- 

-+1. 

i'  = 

=  {V~ 

~1.\5- 

=i-i'=i. 

t^-- 

=  (V~ 

ITi  \fi  = 

=i-i'=i- 

^  = 

■■i'=- 

%'-- 

=(y~ 

^1)'^ 

=i-f>=- 

-i. 

i^  = 

-(!'- 

3~1  \H  - 

^i-i'=- 

-i'- 

=  4-1. 

i^-- 

-iv~ 

-1V  = 

=  i-i^  =  i. 

SURDS  AND  IMAGINARY  NUMBERS  225 

Hence,  the  successim  powers  of  i  have  only  the  four  different 
values:  i    —  i,  —  i,  — \    — i,  + 1^  rej^eating  in  regular  order. 

Since  *"^=— 1,  and  since  (  — 1)^  or  to  any  even  power,  is  1, 
then 

e-+^  =  i*n.f    =-1; 

Thus,  *25^t*«+'  =  i;    ^l2^^='•*=l;     iP=i}-'+^=-i. 

154.  Addition  and  subtraction  of  imaginary  numbers. 

Imaginary  numbers  to  be  added  or  subtracted  must  first  be 
reduced  to  the  tupical  fonn^  and  then  combined  as  in  the  addi- 
tion and  subtraction  of  surds. 


Example  1 .     Simphf y  1/  - 4  +  V -9  +  i/-25-i/  — 16. 

=  (2  +  3  +  5-4)j 
=  6^. 


EXERCISE    70. 

Reduce  to  the  typical  form : 

1.  V~=m.  5.  y-^^.  9.  y~=^K 

2.  V^=ms.  6.  i/:=r|f.  10.  i/_i6a;y. 

3.  |/^=^25.  7.  1/^=^7. 


_  11    ,/     9^"^^ 

4.  1/-^.  8.  |/-98.  '    K       49a;y^* 

12.  |/-81(£c-//)^  13.  i/-a^-2a^-6^ 

14.  |/24a;y-9aj-'-16yl 
Simplify : 

15.  yZIg-^    114+^/^307        16.  1/-^+]/ =^100"- 1/^=49: 
15 


226  ALGEBRA 

17.   i/^:^-|/~4-i/^^6.        18.  i/— 8  +  ^rrio  +  |/~7. 

19.  a|/^=^2+-i/^^  +  3i/^:^. 


20.  2xi/—xY-i'Si/y-4x'i/'. 

21.  Add  x  +  2i/^/^=ri  and  2x~i/i/^^l. 

22.  Add  10-|/^:^9  and  6  +  i/^=^. 

23.  Add  a  +  b+  ^/-^h  and  a^-h-  V~h. 

24.  From  a  +  \/^-b  take  a  —  \/^^b. 


26.  From  3aj — 2y\/  —  1  take  a;  +  \/  —  9yl 

155.  Multiplication  of  imaginary  numbers.  Imaginary  num- 
bers to  be  multiplied  should  first  be  reduced  to  the  typical 
form.  Then  proceed  as  in  tlie  multiplication  of  surds,  ob- 
taining the  product  of  the  factors  V^-^\  by  use  of  §  153. 

Example  1.     Multiply  v— 4  by  ]/  — 9. 


Example  2.     Simplify  V -16V —25V —i'^. 

l/^;^l/^^l/-49=4l/^-5l/^.7l/^  =  140i-^=-140j.  ^ 
Example  3.     Multiply  i/  — 3  +  i/  — 5  by  i/— 2— i/— 4. 

l/~i  +  V^=iV^+iV5  ;    1/^-1/^=^/2-2*. 

^V34-^l/5 
^/2-2i 

i'  1/6  + 1'  j/ 10  -2P  }/S-2i'  1/ 5  = 

- 1/6  - 1/10  +  2i/3  +  2i/5. 

156.  Two  complex  expressions  which  differ  only  in  the 
signs  of  the  imaginary  terms  are  said  to  be  conjugate. 

Thus,  a?  +  7/i/— 1  and  x—yi/—l  are  conjugate  expressions. 
The  product  of  tioo  conjugate  complex  expressions  is  real. 
For      {x-\-y\/^-V){x-y\/^=\)=x^—y\-V)==x^-Vy\ 


SURDS  AND  IMAGINARY  NUMBERS  227 

15  7.  Division  of  imaginary  numbers.  Imaginary  numbers  in 
division  should  first  be  reduced  to  the  typical  form.  Then,  in 
general,  multiply  both  dividend  and  divisor  by  such  an  expres- 
sion as  will  make  the  divisor  real  and  rational. 

Example  1.     Divide  6  by  l/^^. 


Example  2.     Divide  ]/  —9  by  y  —25. 


9     31/ -1 


1-25     51-1 

Dividing  both  terms  by  1/  —1,  =^ 

Example  3.     Divide  2  +  1^^  by  3 -  V-Tx. 

2  +  l/^  _  2  +  /1/5  ^  ( 2  +  n/5)(3  +  i]/5 ) 
3-l/^~3-^l/5~(3-^l/5)(3^-^V5) 

_6  +  5/l/5  +  5/^     l  +  5zi/5     1       5iVy 
-        9_5i-^         -        14       -14"^     14    • 


EXERCISE  71 

Simplify : 


1.  i/=25i/^^.  3.  |/^=T00i/-49. 

2.  i/^=:9]/^=l6.  4.  y^^^i/^^i-Oe. 

5.  i/^=25i/^4i/^^36i/^=49. 


6.  i/^=^l6i/"=^|/-25i/-81i/-4. 

7.  -i/=7^V^=^^^  ^'  (l  +  l/^=4)(l-l/^. 

8.  -i/ir^^(_v/^.  10.  (3  +  i/"=2)(2-i/^=^). 

11.  (1/^=2  +  1 /=3)(l/'=^-l/^). 

12.  (2i/^=^  +  3|/^(3i/^=^-2i/^^). 


228  ALGEBRA 

13.  (^y^^  +  S]/^=l.)(2i/^^b-i'^). 

14.  (1/^=^  +  1/^(1/ -^-l/^. 

15.  {xy'^-yy^)(x}/^=^+(/i/^^). 

—  b}/—x^ 
2i/  —a; 
2 
^^-  1  +  1/^2' 

l  +  i/^^F 

30.  ^, -==y 

1-V  -2 

3-|/==~5 

^^•sTi;^- 


16. 

5 
1/-1 

3 

17. 

2i/-l 

18. 

10 
1/-4 

19. 

21 

l/=9* 

20. 

a; 

1/-X 

91 

4,/-l 

22. 
23. 

24. 

1/-16 

1/-9 
1/-100 

i/'^2r 

—  X 

y'-9x' 

25. 
26. 

2i/-25x* 

3i/  — 16a;^ 
1/-16 

1/-25 

07 

l/-25a;« 

32. 

6  +  1/- 

-3 

3-1/= 
1 

=^ 

33. 

x+^ 

5  +  2i/ 

/  -y 

-5 

■  |/-4  '^"  i/  — 49a;' 

34.   '   ^      '  -^'  35. 

1/-5-1/-7  .T-i    -2/ 

158.  Geometric  representation  of  imaginary  numbers.  Ac- 
cording to  the  method  of  Chapter  XIII  all  real  Clumbers  may 
be  represented  by  distances  measured  along  the  line  XX' 
from  the  point  0,  positive  numbers  by  distances  to  the  right, 
and  negative  numbers  by  distances  to  the  left. 

Let  us  consider  any  positive  number,  say  2,  represented  by 
OA.  By  revolving  OA  counter  clockwise  about  the  point  0 
it  may  be  made  to  take  the  position  of  OB^  0  C,  OD^  or  any 
other  line  drawn  through  O. 

Now,  multiplying  2  by  y^A  gives  2y  ^^1.  Multiplying 
again  by    v— 1  gives  —2,  which  is  represented  by    OC  in 


SURDS  AND  IMAGINARY  NUMBERS 


229 


the  figure.     Hence,  multiplying  2  twice  by  |/^^1  gives  the 
same  result  as  revolving  OA  into  the  position  OC  ;  i.  e.,  reverses 

the   direction  of  the  line 
'  which  represents  the  mul- 

tiplicand 2.  Since  mul- 
tiplying twice  by  i/^ 
revolves  the  line  OA  into 
the  position  0  C\  multiply- 
j^'  ing  once  by  |/  — 1  may  be 
interpreted  as  revolving 
OA  through  only  half  of 
the  distance,  i.  6.,  into, 
the  position  OB.  Mul- 
tiplying three  times  would 
revolve  it  into  the  posi- 
tion OD.  And  multiply- 
ing four  times  would  re- 
volve it  into  the  position  OA  again.  But  multiplying  once 
gives  2|/  —  1 ;  multiplying  twice  gives  —2  ;  multiplying  three 
times  gives  —  2]/  — 1;  multiplying  four  times  gives  2. 
Hence,  2|/  — 1  is  represented  by  OB^  and  —  2]/^^l  is  repre- 
sented by  OD.  The  same  interpretation  would  hold  in  case 
of  any  other  real  number  as  well  as  in  the  case  of  2.  Hence 
the  following  principle : 

If  all  real  numbers  are  represented  hy  distances  measured 
along  the  line  XX'  from,  0,  positive  numbers  to  the  right  and 
negative  numbers  to  the  left,,  then  all  imaginary  numbers  will 
be  represented  by  distances  7neasured from  0  along  the  line  YY' 
perpendicular  to  XX\  those  with  positive  coefficients  above  0,  and 
those  with  negative  coefficients  below. 


CHAPTER  XV. 
QUADRATIC  EQUATIONS— ONE  UNKNOWN  NUMBER. 

159.  Quadratic  equations.  Any  quadratic  equation^  or 
equation  of  the  second  degree^  in  an  unknown  number  x  must 
have  terms  in  x'^,  and  may  liave  terms  in  x,  and  terms  free  ofx, 
but  710  other  kinds  of  terms. 

A  quadratic  equation  that  contains  a  term  of  t\\Q  first  degree 
in  the  unknown  number  is  called  a  CDinplete  quadratic  equation. 
A  quadratic  equation  that  does  not  contain  a  term  of  the  first 
degree  in  the  unknown  number  is  called  a  pure  quadratic  equa- 
tion. 

Thus,  3x^  — 5x=6  is  a  complete  quadratic  equation.  And 
5x^—4=0  is  apui^e  quadratic  equation. 

By  grouping  terms,  any  complete  quadratic  equation  can  be 
written  in  the  type  form 

in  which  x  is  the  tmknown  7ium,her,  and  A,  7?,  and  C  are  hnovin 
numbers  having  any  finite  values. 

Any  pure  quadratic  equation  can  be  written  in  the  type 
form 

Ax'^B  =  0, 

in  ichich  x  is  the  iinhnovin  7ium,her. 

160.  To  solve  a  pure  quadratic  equation.  Any  pure  quadratic 
equation  can  be  solved  like  the  following  : 

Example  1.     Solve  23(^  —  12=2>Q-x\ 

Adding  a?^  + 12,  2x^-\-  x' = 36  + 12.  (Authority  ?) 

Uniting  terms,  3a!^=48. 

230 


QUADRATIC  EQUATIONS— ONE  UNKNOWN  NUMBER    231 

Dividing  by  3,  x^=16.  (Authority  ?) 

Hence,  x  must  be  a  number  whose  square  equals  16  ;  i.  e., 

x=i/T6 
=4  or  —4. 
This  may  be  written  x=  ±  4. 

2 

EXx\MPLE  2.     Solve  <^  +  l=~zrTv 

Multiplying  both  members  by  'a — I ,       a'^  —  l=2. 
Adding  1,  0^=2  +  1. 

Uniting  terms,  a^=3. 

Hence  a  must  be  a  number  whose  square  equals  3  ;  i.  e., 

a=  ±  I  3 
The  solutions,  or  roots,  are  yS  and  —  ]/3. 

These  examples  illustrate  the  following  rule  : 

(1)  Clear  the  equation  of  fractions^  if  necessary,  and  rentoce 
all  signs  of  grouping. 

(2)  Transform  the  equation  into  an  equivalent  equation 
having  all  the  unknown  numbers  in  the  first  member ,  and  all 
terms  free  of  the  unknoimi  yiumber  in  the  second  'member. 

(3)  Unite  like  tertns. 

(4)  Divide  both  members  by  the  coefficient  of  the  unJcnown 
number .^  thus  giving  an  equation  of  the  form  x'^=A,  where  x  is 
the  unknoimi  number. 

{5)  Extract  the  square  root  of  both  ynembers  of  tJie  resulting 
equation.^  attaching  the  double  sign,  ±,  to  the  second  member. 

EXERCISE  73. 

Solve  for  the  general  number  : 

1.  7a.'^-28  =  0.  ■  3.  r^  =  845-4rl 

2,  10.r^-150-4a;l  4.  «(rt  +  4)  =  4a  +  49. 


232  ALGEBRA 

e.  Sx'  =  2x{x-b)  +  10{x  +  2).^      '      ^  ^ 

^  _^  16.  ^-^  +  ^J^1. 


7. 


125     X 


3^4  17    3a;^-7     a;-l_a;  +  l 

8.    I g^-  '       CC^— 1       £C+1       £C— 1 

9       ^    -''/-^  18.  ^^-4l=l. 

^-  ^rn — 6^"  6    "+1 

10  ?^=i.  19.  r^.-^^r 
•     8  6s+l       2  +  0 

11  »^'  +  8     o  on   4a3  +  l_8^-19 
11-  -2""^^-  ^"'  3^=4"7^=24- 

..    9x^-l_3  21    %  +  ^-     1^ 

t/l-i.  — 3  ^.  -61.     2  +  y     2y  +  3' 

/l3.  -^^4.  22.  4^+i  =  l^. 
^)^  — 1  5         3a;— 2 

J^_l_9  3s  +  6     2s  +  5 


/^ 


2/     y^     4*  ""•  22-6     32-6' 

^  '  a;  +  2 

25.  4£z!=^.  27.  (y  +  i)(y-4)  =  2. 


5£e- 

-4 

4x- 

-5' 

2a; 

1  + 

x' 

3 

5"~ 

2x- 

f3" 

-5" 

X^  +  X-^1  £C-  — £C+1 

x—l  x+l 

161.  Ill  the  following  sections  three  methods  of  solving  a 
complete  quadratic  equation  are  discussed :  (1)  by  factoring  ; 
(2)  by  completing  the  square  ;  (3)  by  the  use  of  a  formula. 

Note. — The  factoring  method,  it  will  be  found,  may  be  used  to  solve 
an  equation  of  any  degree  higher  than  the  first  in  whicli  rational  fac- 
tors can  be  found.  But,  since  comparatively  few  expressions  have 
rational  factors,  the  method  of  solving  equations  by  f^,ctonng  may 
be  used  in  only  a  limited  number  of  cases. 


QUADRATIC  EQUATIONS— ONE  UNKNOWN  NUMBER    233 
162.  To  solve  complete  quadratic  equations  by  factoring. 

This  method  is  based  upon  the  principle,  that,  in  general^  a 
product  is  zero  lohen  one^  or  tnore^  of  its  factors  is  zero^  and 
not  otherwise. 

Thus,  in  general,  ah  is  0  if  a  is  0,  or  if  h  is  0.  Also  if  ah  is  0, 
then  either  a  or  6  must  be  0. 

Note. — This  principle  fails  if  one  of  the  numbers  is  indefinitely  great 
at  the  same  time  that  another  factor  is  0.     See  (4)  §  219. 
For  example,  the  product 

8 


(^-^>{^) 


Now  when  x=2  the  product  is  equal  to  |,  or  2,  and  not  to  0,  although 
the  factor  x — 2  becames  0. 

Example  1.     Solve  x^  +  Q  =  5x. 

Subtracting         5x,   x^—5x  +  Q=0. 

Factoring,  {x—S){x—2)=0. 

This  equation  is  satisfied  by  a  value  of  x  that  Avill  make  either 
factor  of  the  first  member  0.  Such  values  of  x  are  obtained  by 
equating  each  factor  to  0,  and  solving  the  resulting  equations. 

Equating  each  factor  to  0,  we  have 

a*— 3=0  ;  whence,  x=S. 
0?— 2=0  ;  whence,  x=2. 

Check.     3'  +  6 = 5  •  3  and  2'  +  6 = 5  •  2,  identities. 

Example  2.     Solve  Qx^=x+2. 

Adding  -x-2,  Qx^-x-2=0. 

Factoring,  (3j;-2)(2ir  4- 1)=0. 

Equating  each  factor  to  0, 

3x—2=0  ;  whence,  a?=|; 

2^^+1=0  ;  whence,  x=—}. 
Let  the  student  check  the  two  solutions. 


231  ALGEBRA 

To  solve  an  equation  by  factoring  we  evidently  have  the 
following  rule : 

(1)  Transform    the    equation   into    an    equivalent   equation 
having  all  of  its  terms  in  the  first  member. 

(2)  Then  find  the  rational  factors  of  the  first  member. 

(3)  Equate  each  of  these  factors  to  0^  and  solve  the  residting 
equations. 

EXERCISE  73. 

Solve  by  the  factoring  method  : 

1.  x'-1x^Vl  =  ^.      ^-    xll  --    r:_4  ,  33 

2.  i«M-3£c-10  =  0.  A     1     X      ■  XX' 

/3.  a3^  +  8.T  +  7  =  0.        j3    14-??=^:.  ^^'  «'  +  i«=¥- 

x"      x'  22 

22.  1+--Gy  =  0. 

6.  2a;^-5£c  +  2  =  0.      14.  l5a;  +  4=-.  ^ 

X  23    6f^2  =  11^  +  7 

7.  W  +  12.^24.      15-  -+1^1-22..       24.17.^^70.-8. 

^  8.  hx^=l^Ax.  ^^'  6^^  +  7^=-2-     ;J5.  12/-7y  +  l  =  0. 

*   9.  2Lt^  =  10  +  29«!.  ^'''-  4^-^l'=-77.    26.  96  +  22/=.3yl 

10.  6.«'''-ll^=2.  18.  P^-llP--30.27.  10^-3  =  3^^ 

11.  3.^^  +  5i«  =  2.  19.  i«='-3£c-10  =  0.     28.  lO-s^-Ss. 

29.  ;«^- 23a; +  132  =  0.  ^.    32^-4     ^   ,, 

^  34.    ^  ,  ,)-=2g4l. 

30.6-^^4^  =  0.  t' 

21  y+3    '^ 

31.  5r— 8= — .  o       1A         ,  ^oA 

r  „^    3ic— 10     £c  +  120 

OD.    =-i = . 

32.  2/^  =  13y-36.  14  x 

«o      ,  ,  10     ,    1      ^  OP,        1         ^-7  ,   a;  +  24      _ 

33.  .^+-.  +  -=0.  37.  2-:z2-,rxi  +  5^^^=^- 

38.  -^,-^0=2. 


QUADRATIC  EQUATIONS-ONE  UNKNOWN  NUMBER    2S5 

163.  Completing  the  sq[iiare.     From  the  identity 

a?^  -\-^ax-V  a?  =  {x-{-  a)  ^, 

we  have  x^^-^ax-^-a^  as  the  general  form  of  the  perfect 
square  of  a  binomial.  Hence,  if  we  have  given  the  binomial 
x^  +  2a£c,  to  make  it  a  perfect  square,  we  must  add  a^^  ^^e.,  the 
square  of  half  the  coefficient  of  x.  The  process  of  adding  a 
term  to  a  binomial  such  as  a^^  +  2a.T,  in  order  to  form  a  perfect 
square^  is  called  completing  the  square. 

Example  1.  Form  the  perfect  square  whose  first  two  terms 
are  x^  —  V)x. 

/10\  ^ 
Here  we  addf  -^\  ,  ov  25.    This  gives  a?^— 10^  +  25. 

Example  2.  Complete  the  square  of  which  two  terms  are 
x}  4-  3a?. 

Here  half  the  coefficient  of  x  is  |.  The  square  of  |  is  \.  Add- 
ing I,  we  have  x'^  +  3a?  + 1 .     a?'  +  3a? + 1  =  (,r + 1)^ 

Note. — It  would  be  well  here  for  the  student  to  review  §  61. 

164.  Solving  the  complete  quadratic  by  completing  the  square. 

Any  complete  quadratic  equation  whatever  can  be  solved 
by  use  of  the  principle  of  §  163. 

Example  1.     Solve  a?^  +  3x— 10=0. 

Adding  10,  x"-  +  3x- 10. 

Adding  Q)^  to  both  members,  £C^  +  3a?  +  |=— . 

Extracting. the  square  root,  a?  +  |=±2- 

Solving  for  a?,  '■^=—1  ±  I- 

Using  the  +  sign,  a?=  — 1  +  |,  or  2. 

Using  the  —  sign,  a?=— |— |,  or  —5, 
Hence  the  two  roots  are  2  and  —5. 
Let  the  student  check  the  results. 


236  ALGEBRA 

Example  2.     Solve  2x^  —  5x=7. 

Dividing  by  2,  x^—§x=l. 

Adding  the  square  of  half  of  |,  a?^  — |x  +  f  |=|  +  ||,=|^. 

Extracting  the  square  root,       x—^=  ±  |. 

When  £c— 1=1,  x=l.     When  ic— f^— |,  x=  —  l. 

Check  the  solution. 

Examples.     Solve  x^—4:X  +  5=0. 

Adding  —5,  x^  — 4a?=  — 5. 

Adding  square  of  half  4,  x^  —  4x-\-4^=  —  l. 

Extracting  the  square  root,      .r — 2  =  ±  y/  —  1 . 

When  ir— 2=1   ^,  ir=24- 1/^. 
Whenir-2=-i   ^,  x=2-l   ^. 
These  may  be  written  x=2±  ]/  — 1. 
Here  the  two  roots^  or  solutions,  are  complex  numbers. 
Check  the  solution. 

Evidently  to  solve  any  quadratic  equation  we  have  the  fol- 
lowing rule : 

(/)   Reduce  the  equation  to  the  form  Qt?-\-px  =  q. 

(2)  Add  to  each  member  the  square  of  half  the  coefficient  of  x. 

(3)  Extract  the  square  root  of  both  members  of  the  restdting 
equation.,  attaching  the  double  sign  ±  to  the  resulting  second 
member. 

(4)  Solve  the  tico  restdting  linear  equations. 

Observe  that  the  double  sign  might  have  been  placed  before  the 
resulting  first  member  as  well.  The  reason  for  not  doing  so  may 
be  seen  from  the  following  example. 

Let  x^=a^. 

Extracting  the  square  root  of  both  members,  ±_x=±_a,  that  is 
x=-\-a  or  —a  and  — a?=  +a  or  —a. 

Now  if  in  this  second  equation  we  divide  by  —  1,  we  have  x=  —a, 
or  +  a,  the  values  in  the  first.  That  is,  .t=  ±  a  is  the  same  as  —x= 
±a;  hence  we  use  the  double  sign  before  the  second  member  only. 


QUADRATIC  EQUATIONS— ONE  UNKNOWN  NUMBER    237 


EXERCISE    74. 


3.  x' 

4.  x' 


Solve  by  completing  the  square  : 

1.  x}-%x  +  ^  =  (). 

2.  x'-  +  4:X-=V2,. 

7.  x'-^x^^. 

8.  a;^-9£c=22.  ^  4  + a; 

9.  x'-lbx^U.  31.   -Sit-^ 


10 


.   X' 


12a; +  20  =  0. 


11.  2a;2  +  i«=l. 

12.  "Ix'-bx^^. 

13.  6a;^  +  6  =  13a;. 

14.  4a;2-lla;=3. 

16.  10a;2-29£c  +  10  =  0. 

16.  3a;^  =  17a;  +  28. 

17.  2a;^  +  3a;=5. 

18.  f>x'-x=^l. 

19.  9ic^  +  3£c  +  18  =  0. 

20.  12£c^-14a;  +  3  =  0. 

21.  0^=^  +  80;  + 21  =  0. 

22.  aj^  +  6aj+ll  =  0. 

23.  ^x^-'^Q  =  Qx, 

24.  i«^  +  6a;  +  25  =  0. 

25.  16a;^-96«=1792. 

26.  ic^-8a;=-15. 

27.  x'  +  Ux=-^h. 

28.  39icH96  =  51a;='-96. 
1  2  18 


29. 


ic4  1     1— JC     4ic— r 


14a;  +  40=-0.     5.  a;^-lla;  +  30  =  0. 

6.  £c-  f  5ic  =  14. 

5  _8 
4:-x     3" 


ic=6. 
30 


36.r-105  =  0. 


32. 


20, 


42|. 


33.  6.3+?^:::^=44. 


34. 


7a.-=7(fc  +  3)  +  4. 


X—^        X+4: 

36.  ^5-,^7-2i. 


36. 


37.  , 


1 


£C+1       X—1 

2 


=  1. 


38.  3 


2a!-3  '  a; 
1 


+  2  =  0. 


39. 


40. 


icH  2     a;-2' 

£C  — 2       37  —  3 


JK-l 

a;  +  2     4-£c     7 


0. 


ic-1     2aj         3 

41.  (a;-5)^  +  (ic-10)^  =  37. 

42.  ^^  +  ^  =0. 


43. 


if  +  9  '  a;-l 


•  + 


x'-\  '  2a;-2     4' 


or  THi    ^ 


238  ALGEBRA 

44    __i L  =  _l  45    _A  .  _1=11 

**•  £c  +  2     x-'l     l-x  ^'''  x-l^x-3      2* 

46.  {4:i^xy  +  (x-4y  =  Q(x-4:)(x  +  4). 

2.;^-l     a;— iSa;— 10  a;—  4  _   a—  4      11 

*'•  "^^^£^-2""  a;-   3'  *^'  2a3-12     3.t-16~^' 

165.  Solving  a  quadratic  equation  by  the  formula.     Since  any 
complete  quadratic  equation  may  be  thrown  into  the  type  form 

the  solutions  of  this  equation  will  give  7i  formula  by  which  the 
solutions  of  any  particular  quadratic  equation  may  be  written 
down  at  once. 

Solve  Ax^  +  Bx  +  C=  0  by  completing  the  square. 


Dividing  by  A^ 

-'+a-+3=«- 

Adding  -^, 

Completing  the 

square, 

^., ,  ^  ^  B'-    B^    a 

^+yl^  +  4.P     4.4^     A' 

,^B    ,     B'     B'-4A0 

Extractmg  square  root,    ^  +  w~i=  ±|/  - — j-p 


Solvnig  for  x,  ^^~~2A^ 2A —  ' 


/        -B±yB'-4A(y 
or  ^J  x= 271 -• 

.p,    ,.               w-       •                      -i?+i/i5^-4^6^ 
1  hat  IS,  one  solution  is  2~4~~ — ~ ' 


ifi.fl-                                    -B-rB'-4A0 
and  the  other  is 0-3 • 


QUADRATIC  EQUATIONS— ONE  UNKNOWN  NUMBER     239 

Now,  by  replacing  A,  B,  and  C  in  this  formula  by  the  par- 
ticular values  which  they  have  in  any  given  equation,  the 
solutions  of  any  quadratic  may  be  written  out.  Thus,  the  long 
process  of  completing  the  square,  etc.,  in  every  equation  may 
be  avoided.  Hence,  the  student  should  master  this  formula  and 
tise  it  in  all  future  VDorh  where  the  quadratic  eannot  he  factored 
at  sight.  He  should  be  able,  hoAvever,  to  solve  any  quadratic  by 
completing  the  square,  and  in  this  way  to  derive  the  formida. 

Example  1.     Solve  Sa?^— 4ic=15. 

Written  in  the  form  Ax'^  +  Bx-\-C=0,  this  becomes 

Here  A=3,  5= -4,  C=-15. 

„                                                                  4  ±1/16  +  180 
Hence,  x= = — ^ 

^4±14 
6     • 

'  18 
Using  the  +  sign,  ^~"~6'  ^^  ^' 

Usmg  the  —  sign,  x= g,  or  —  o- 

Let  the  student  check  the  result. 
Example  2.     Solve  4ic2  +  2=— Sa:*. 
Adding  3x,  we  have  4^?^  + 3a? +2=0. 
Here,  A=4,  5=3,  C=2. 


-3  ±1/9-32 
Hence,  x= ^-g 

_-3±j/^ 

~  8 


-3  +  1-23             -3-1/-23 
= '-^ or ' 

Let  the  student  check  the  result. 


8  ""^  8 


240  ALGEBRA 

EXERCISE  75. 

Solve  by  use  of  the  formula : 

1.  a;''-3a;-10.  8.  Qx'  +  Q  =  Ux.       J,5,  2b'-7b  +  Q  =  0. 

2.  aj''  +  10ic  +  21  =  0.     9.  10x'  +  U  =  ^9x.     16.  W  =  Qk-b. 

3.  2x'-i^x=Q.  10.  8£c=^-65£c+8  =  0.   ^^    21b'-b  =  4: 

4.  ^x'  +  bx=2}  11.  2«^  +  3a  +  4  =  0. 

1ft    q^ o'2z=fi 

5.  6i«^  +  a;-5  =  0.       12.  ba'-2a  +  7  =  0.   /    '  ' 

6.  7a;^=50a!-7.         13.  15y^-y  +  3  =  0.     l^.  12  +  a;  =  lla;l 

7.  4x'-}:x=b.  14.  y'-8y  +  4  =  0.       20.  l-a^-Sa^O. 

21    ?^^±I 

y      2 

22.  — ^~^=0. 

1-t        3 

23.  i— i  =2. 
24   ^-l_.9-+l_i 

26.  l+v  =  li 

26.  r  +  3  =  -^. 

7*— 1 

27.  #-i=3-i. 

28    2^  +  3_  «^  +  3 
•     x-2     2x-r 


30.  (3a;-l)2-7i«. 
^,       ^„     3a;^-4 

3^-  «^+2=2^-q^r 


32. 

(5.T  +  2)2 -(2cc- 3)^  =  0, 

33. 

l-(2x-iy  =  0. 

34. 

10v'  =  bv. 

35. 

Qx-m=^x\ 

36. 

1-1-221 

37. 

2a!  +  |  =  2iK^ 

38. 

^x'  =  42-6x. 

39. 

■+"m- 

>40. 

„=j-.. 

41. 

a-a'  =  0. 

42. 

3     _    6            8 
x—4     x  —  b     x  —  S' 

43. 

x-2     x  +  2     10a;-8 

x  +  2  '  x-2       4-x' 

44. 

,            l  +  2a;2 

1— CC  =  -v— . 

5  +  a; 

0. 


QUADRATIC  EQUATIONS— ONE  UNKNOWN  NUMBER     241 

45  2  l-xx+l  -g    8  +  a;     4-2a;_8 

5(a;-2)     a;  +  2     a;^-4'  *  8-ic  +  4  +  2a;~3' 

-c    7  — 3£c  ,     7-£c     2x  +  S  ^^    ^       Sx-S     ^    ,  dx-6 

40.  -j — 0-4-0 — TTJ^^'^i rt-  50.  ox— ii  =  2x-{- — o — 

4— 3a;     2a;  +  2     Ax— 2  x—S  2 

47    1     ,      y    _  y  +  3  51       6        2 

^^-  g  +  4(^)  =  2^-  ^^-  ^^-^=14- 

2^4 
Hmf. — Consider  (x — 1)    the  unknown  and  this  is  a  quadratic  in 

166.  The  discriminant.     In  the  use  of  the  formula  of  §  165, 

x= k-i 5  it  was  observed  that  the  character  of  tUe 

2A 

roots  depended  upon  the  part  under  the  radical  sign.     This 

quantity,  B'^—AAC,  for  that  reason,  is  called  the  discriminant  of 

the  quadratic* 

Observing  the  formula,  we  see  that 

(i)    When  B'—JfAG  is  a  perfect  square,  the  roots  are  real-, 
rational,  and  unequal. 

(2)    When  B'—Jf^AC  is  equal  to  zero,  the  roots  are  equal. 

{3)    When  B-—j^AC  is  positive  but  not  a  perfect  square  the 
roots  are  real  and  conjugate  surds. 

(4)    When  B^ — J^AG  is  negative  the  roots  are  conjugate  com- 
plex numbers^ 


*  It   is  assumed    in  principles  (1),   (2),   (3),  and    (4)    that  the   co- 
efficients ^,  J5,  and  Cof  the  general  equation  Ax'^-{-Bx^rC=^  are  all 
real  numbers.     The  principles  may  or  may  not  be  true  when  one  or 
more  of  the  coefficients  are  imaginary. 
16 


242  ALGEBRA 

From  these  observations  we  can  determine  the  nature  of  the 
roots  of  any  quadratic  equation  without  solving  it. 

Example  1.     Determine  the  nature  of  the  roots  of  a?^—5a^  + 6=0. 

Here-B=  — 5,  A=l   and   C=Q  ;  hence,  B'— 4AC=1,  a  perfect 
square;  hence,  the  roots  are  real,  rational,  and  unequal. 

Example  2.    Determine  the  nature  of  the  roots  of  x^-\-x-i-3=0. 
Here  JB^— 4AC=  — 11,  hence  the  roots  are  conjugate  complex 
numbers. 

Without  solving  tell  the  nature  of  the  roots  of  Examples  1  to 
20  in  Exercise  76. 

167.  The  relation  of  the  roots  to  the  coefficients.     By  dividing 

both  members  of  the  general  quadratic,  Ax^-}'JSx+  0=0,  by  A, 

B       C 
we  have  x^  +  -ix-\--j=0,  which  is  of  the  form  x}+px  +  q  =  0, 

where  ^  and  q  may  have  any  finite  values,  integral  or  fractional. 

Solving  9?H7>a3  +  5'  =  0  by  the  formula  Ave  have 

_  —p  ±  y'jf  —  4:q 
X  2  • 

Calling  one  root  r^  and  the  other  r^,  we  have 
_—p  +  l/2f—4q 


and  r.=^I^PJZ^, 

Adding  these  we  have 

r^  +  r^=—p. 
Multiplying,  we  have 


.^^^^  Tzi^+vVzlil  X  r-i^-ri^'-4(y1 


p'-(p'-4:q) 
4 


QUADRATIC  EQUATIONS— ONE  UNKNOWN  NUMBER     ^43 

From  these  relations  we  can  state  the  following  principles : 
When  an  equation  is  throicn  into  the  form  x^-\- px-\-q  =  0  ; 
(1)    The  sum  of  the  roots  is  the  coefficient  of  x  with  its  sign 

changed  ; 

{2)    The  product  of  the  roots  is  tlie  term  free  from  x* 
From  these  relations  we  may  form  equations  with  given 

roots. 

For  example,  form  an  equation  whose  roots  are  5  and  3. 
Since  the  coefficient  of  a?  is  —  (5  +  3)  and  the  term  free  from  x  is 
3  •  5 ,  the  required  equation  is  x^ — 8.r  + 1 5 = 0 . 

EXERCISE  76. 

Write  the  quadratic  equations  whose  roots  are : 

1.  6,  -5.  ^   _3^  6.  2,  -f. 

2.  -2,  7.  *  -^'  7.  -4,  -6. 

3.  \.  5.  i,  -|.  8.   -f,  -4. 

9.  i,  1.  10.  -i   -i 

11.  By  the  use  of  principles  1  and  2,  §  167,  show  what  the 
signs  of  the  roots  of  ic^  — 11.^  +  24  =  0  must  be. 

12.  What  are  the  signs  of  the  roots  of  £c'  +  5a;— 24==0?     Of 
a;'^-7i^-18  =  0? 

13.  For  what  value  of  a  will  the  equation  aic^  +  3.7;— 5  =  0 
have  equal  roots  ? 

14.  For  what  values  of  m  will  the  equation  2a;^  +  mic  +  32  =  0 
have  equal  roots  ? 

15.  For  what  values  of  c  will  3a;'^— 2£c  +  c=0  have  real  roots  ? 
For  what  value  of  c  will  the  roots  be  equal  ? 

*  The  term  of  an  equation  free  from  the  unknown  is  called  the 
absolute  term. 


-2h±y4b'  +  16x 

-8 

-2b±2]/b'+4x 

-8 

24:4  ALGEBRA 

168.  Solving  a  quadrate  formula.  In  §  119  we  solved  some 
formulae  which  were  linear  with  respect  to  certain  general  nmn- 
bers  involved.  By  the  methods  of  the  preceding  sections, 
formulae  which  are  of  the  second  degree  with  respect  to  a 
certain  general  number  may  now  be  solved  for  that  number. 

Example  1.     Solve    x=2a{2a—h)  for  a. 
Removing  the  sign  of  grouping,  x=4a^—2ah. 

Arranging  in  type  form,  —  4a^  +  2a6-f  a?=0. 
Here  A=-4,  B=2b,  C=x. 

Hence,  by  §  165,  a= 

Example  2.     Solve    1=-^  for  f . 

Multiplying  by  c^  Ic^—af^. 
Hence,  —at^=—lc^. 

Dividing  by  —a,       ^^=77' 

Hence,  f=±l/— ,    or  ±c|/Z,   or  ±^y^W, 

^      a  f^     a  a^ 

EXEBCISE  77. 

1.  ax'^—c=Q.     Solve  for  x. 
.  2.  aW-a'=^.     Solve  for  h. 

3.  ah  =  a'^—A.     Solve  for  a. 

4.  2ic^  —  3a'  =  bax.     Solve  for  a;,  and  for  a. 

5.  x'^^-2a^  =  Mx.     Solve  for  aj,  and  for  a. 

6.  a3^  +  2«£c=^-  +  2«A.     Solve  for  5,  and  for  x. 

7.  £c'  +  ra;  +  s  =  0.     Solve  for  a;. 

8.  mx^-\-nx^t=^.     Solve  for  aj. 


QUADRATIC  EQUATIONS— ONE  UNKNOWN  NUMBER     245 

9.  a?+-=aH—    Solve  for  a. 

'  a  X 

The  following  formulse  express  important  laws  of  physics. 

10.  5=^/7^.     Solve  for  ^.  14.  i^=-^.     Solve  for  f?. 

11.  /='^1    Solve  for  t.  16.  E=\mv'.     Solve  for  v. 

12.  F=^  —  .     Solve  for  v.  16.  r=h-^.     Solve  for  d. 

r  d- 

13.  F=^.    Solve  for  v.  17.  s^vt  +\gt\     Solve  for  t. 

18.  ?i^=-T2--T.     Solve  for  ,9,  /,  and  n. 

169.  Problems  which  lead  to  quadratic  equations. 

Some  problems  can  be  solved  by  the  use  of  quadratic  equa- 
tions. It  has  been  seen  in  §  166  that  the  solutions,  or  roots,  of 
a  quadratic  equation  may  be  whole  numbers  or  fractions, 
positive  or  negative,  rational  numbers  or  surds,  real  numbers 
or  imaginary  numbers,  depending  upon  the  relation  among  the 
coefficients.  A  problem  may  by  its  nature  require  for  its 
solution  a  certain  kind  of  number.  For  example,  a  certain 
problem  might  require  for  its  solution  real  numbers.  Now,  if, 
by  solving  the  equation,  imaginary  solutions  are  obtained,  they 
must  be  discarded.  Plence,  so7ne  of  the  solutions  of  ayi  equation 
may  satisfy  the  equation  and  yet  not  satisfy  all  of  the  requirements 
of  the  problem.  It  is  necessary,  therefore,  in  solving  problems 
by  the  use  of  equations,  to  examine  the  solutions  to  see  if  all  of 
them  satisfy  the  requirements  of  the  problems. 

Example  1.  Sixty-four  times  the  number  of  students  in  a 
class  exceeds  3  times  the  square  of  the  number  by  21.  Find  the 
number  of  students. 


246  ALGEBRA 

Let         x=  number  of  students. 
Then  Ux=3x^  +  21. 
Solving,  we  obtain  x=21,  or  x=^. 

Evidently  the  problem  requires  for  answer  a  ivhole  number. 
Hence,  the  solution  x=:^  must  be  discarded  ;   and  the  required 
number  of  students  is  21. 

Example  2.  A  man  walks  25  miles  at  a  uniform  rate.  If  he 
had  walked  f  of  a  mile  per  hour  faster,  the  journey  would  have 
taken  2  hours  less.     Find  the  rate  of  his  walking. 

Let  x=  number  of  miles  traveled  per  hour. 

Then     —     =  number  of  hours  required  for  the  journey. 

25 

^nd      ——5—  number  of  hours  the  journey  would  have  re- 

x-j-  ^ 

quired  had  he  walked  |  mile  per  hour  faster. 

25     25 
Hence,  ^=;F4:^  +  2. 

.  Solving,  we  get  x=2l,  or  — 3^. 

The  rate  must  be  an  arithmetical  number.  Hence,  the  solution 
a?=— 3|  must  be  discarded  ;  and  the  required  rate  is  2^  miles  per 
hour. 

Example  3.  Find  the  real  number  whose  square  increased  by 
32  equals  8  times  the  number. 

Let  x=  the  number. 

Then,  £cH32=8iC. 

Solving,  we  get 

a?=4  +  4l/^  and  4— 4|/^. 
These  expressions  are  not  real. 
Hence  the  problem  is  impossible. 


QUADRATIC  EQUATIONS— ONE  UNKNOWN  NUMBER      247 
EXERCISE  78. 

By  the  use  of  quadratic  equations  solve  the  following  prob- 
lems. 

1.  Find  two  arithmetical  numbers  whose  difference  is  7  and 
product  330. 

2.  Divide  31  into  two  parts  the  sum  of  the  squares  of  which 
is  541. 

3.  Find  two  arithmetical  numbers  whose  sum  is  50  and 
product  336. 

4.  One  of  two  numbers  exceeds  25  by  as  much  as  25  exceeds 
the  other,  and  their  product  is  561.     What  are  the  numbers  ? 

5.  Find  two  consecutive  integers  whose  product  is  5852. 

6.  Find  two  consecutive  whole  numbers  the  sum  of  whose 
squares  is  313. 

7.  The  difference  between  two  numbers  is  10,  and  the  sum  of 
their  squares  is  212.     Find  the  numbers. 

8.  The  denominator  of  a  certain  fraction  is  5  more  than  the 
numerator.  If  the  fraction  be  added  to  the  fraction  inverted, 
the  sum  will  be  2J|.     Find  the  fraction. 

9.  If  64  be  divided  by  a  certain  number,  and  the  same  num- 
ber be  divided  by  2,  the  second  quotient  will  exceed  the  first 
quotient  by  4.     What  is  the  number  ? 

10.  A  pupil  was  to  divide  12  by  a  certain  number,  but  by 
mistake  he  subtracted  the  number  from  12.  His  result  was  5 
too  great.     Find  the  number. 

11.  One-half  a  number  plus  the  square  of  the  number  is  68. 
What  is  the  number  ? 

12.  The  side  of  one  square  exceeds  that  of  another  by  3 
inches,  and  its  area  exceeds  twice  the  area  of  the  other  by  17 
square  inches.     Find  the  lengths  of  their  sides, 


248  ALGEBRA 

13.  A  rectangular  field  is  96  feet  longer  than  it  is  wide,  and 
it  contains  298,000  square  feet.     What  are  its  dimensions  ? 

14.  One  side  of  a  rectangle  is  4  inches  longer  than  the  other, 
and  its  diagonal  is  20  inches.     How  long  are  the  sides  ? 

15.  A  lawn  25  feet  wide  and  40  feet  long  has  a  brick  walk 
of  uniform  width  around  it.  The  area  of  the  walk  is  750 
square  feet.     Find  its  width. 

16.  There  are  two  square  lots  ;  the  side  of  one  is  42  feet 
longer  than  the  side  of  the  other.  The  two  together  contain 
2146  square  yards.     What  are  their  dimensions  ? 

17.  A  floor  can  be  paved  with  200  square  tiles  of  a  certain 
size  ;  if  each  tile  were  one  inch  shorter  each  way,  it  would  re- 
quire 288  tiles.     Find  the  size  of  each  tile. 

18.  The  printed  portion  of  the  page  of  a  book  is  4  inches 
wide  and  6  inches  long.  How  wide  must  the  margin  be  in 
order  that  the  whole  page  shall  contain  48  square  inches  ? 

19.  In  the  center  of  a  rectangular  room  is  a  rug  9  feet  by  12 
feet ;  around  this  is  a  border  of  uniform  width.  The  area  of 
the  floor  is  208  square  feet.     What  is  the  width  of  the  border  ? 

20.  The  length  of  a  rectangle  is  6  inches  greater  than  its 
width  ;  and  if  its  width  be  doubled  and  its  length  diminished  by 
3  inches,  the  area  will  be  increased  by  36  square  inches.  What 
are  the  dimensions  ? 

21.  The  perimeter  of  a  rectangular  field  is  184  rods,  and  the 
field  contains  12  A.     What  are  its  dimensions  ? 

22.  Two  men  start  at  the  same  time  from  the  intersection  of 
two  roads,  one  driving  south  at  the  rate  of  3  miles  an  hour, 
and  the  other  west  at  the  rate  of  4  miles  an  hour.  In  how 
many  hours  will  they  be  25  miles  apart  ? 

23.  Two  trains  are  100  miles  apart  on  perpendicular  roads, 
and  are  running  toward  the  same  crossing.     One  train  runs  10 


QUADRATIC  EQUATIONS— ONE  UNKNOWN  NUMBER      249 

miles  an  hour  faster  than  the  other.     At  what  rates  must  they 
run  if  they  both  reach  the  crossing  in  2  hours  ? 

24.  There  are  two  numbers  whose  difference  is  4.  If  240 
be  divided  by  each  of  them,  the  difference  between  tlie  quo- 
tients will  be  10.     What  are  the  numbers  ? 

25.  There  are  two  arithmetical  numbers  whose  sum  is  33. 
If  36  be  divided  by  the  smaller  number,  and  the  larger  number 
be  divided  by  4,  the  sum  of  the  quotients  will  be  10.  Find  the 
numbers. 

26.  A  man  bought  a  certain  number  of  books  for  $7.50.  If 
he  had  paid  25  cents  apiece  more  for  them,  he  would  have  ob- 
tained 1  fewer  for  the  same  money.     How  many  did  he  buy  ? 

27.  A  man  finds  that  by  increasing  his  speed  1  mile  an  hour 
it  takes  6  hours  less  to  walk  36  miles.  How  fast  does  he 
walk  ? 

28.  A  grocer  paid  $2.70  for  eggs.  Pie  found  that  if  he  had 
paid  3  cents  less  for  a  dozen,  he  would  have  received  3  dozen 
more  for  the  same  sum.  Find  the  price  per  dozen  and  the 
number  of  dozen. 

29.  A  merchant  paid  12160  for  some  carriages,  all  of  the 
same  price.  By  selling  all  but  2  of  them  at  a  profit  of  $36 
each,  he  received  the  amount  he  paid  for  all  of  them.  How 
many  did  he  buy  ? 

30.  A  man  sold  a  lot  for  $1125,  thereby  gaining  i  as  many 
per  cent  as  the  lot  cost  him  dollars.     What  did  the  lot  cost  ? 

31.  In  a  certain  number  of  two  digits  the  tens'  digit  exceeds 
the  ones'  digit  by  2,  and  when  the  number  is  divided  by  the 
sum  of  its  digits,  the  quotient  exceeds  twice  the  ones'  digit  by 
3.     Find  the  number. 

32.  A  tree  was  broken  over  by  a  storm  so  tliat  the  top 
touched  the  ground  50  feet  from  the  foot  of  the  stump.     The 


250  ALGEBRA 

stump  was  |  of  the  height  of  the  tree.     What  was  the  height 
of  the  tree  ? 

33.  A  tree  which  stood  at  the  edge  of  the  bank  of  a  stream 
fell  with  its  top  in  the  water.     The  tree  was  60  feet  high,  the 

\^  bank  on  which  it  stood  was  15  feet  above  the  water  level,  and 
the  body  of  the  tree  passed  under  the  surface  of  the  water  at  a 
point  20  feet  from  the  bank.  What  portion  of  the  tree  was 
under  water? 

34.  The  hypotenuse  of  a  right  triangle  is  4  inches  longer 
than  one  leg  and  two  inches  longer  than  the  other.  Find  the 
sides  of  the  triangle. 

-y  36.  One  pipe  can  fill  a  cistern  in  6  minutes  less  time  than  is 
'required  for  another  pipe  to  fill  it.     The  two  together  can  fill 

it  in  10|^  minutes.     Find  the  time  required  for  each  pipe  alone 

to  fill  the  cistern.    ^  '^"^  .  \    '    ^-    "^ 

36.  One  of  two  pipes  can  fill  a  tank  in  28  minutes,  and  the 
time  required  for  the  other  pipe  to  fill  it  is  19|  minutes  longer 
than  is  required  for  the  two  pipes  together  to  fill  it.  Find  the 
time  required  for  the  two  pipes  together  to  fill  the  tank.     -^    > 

37.  It  would  take  B  four  days  longer  to  do  a  piece  of  work 
than  it  would  take  A  to  do  it.  The  two  together  could  do  it 
in  6|^  days.     In  how  many  days  could  A  alone  do  the  work  ? 

38.  With  John's  help  Henry  could  remove  the  snow  from  a 
piece  of  side- walk  in  6^  minutes  less  time  than  would  be  re- 
quired for  Henry  to  do  it  alone.  John  could  do  it  alone  in  21 
minutes.     In  how  many  minutes  could  Henry  do  it  alone. 

39.  A  could  do  a  piece  of  work  in  2  days  less  time  than 
would  be  required  for  B  to  do  it.  A  works  4  days,  then 
leaves;  and  B  takes  his  place  and  finishes  the  work  in  5 
days  more.     In  what  time  could  each  one  do  it  alone. 

40.  The  sum  of  two  numbers  is  24,  and  the  quotient  of  the 


QUADRATIC  EQUATIONS— ONE  UNKNOWN  NUMBER      251 

less  divided  by  the  greater  is  f|  of  the  quotient  of  the  greater 
divided  by  the  less.     Find  the  numbers. 

41.  Two  trains  on  the  same  road  start  at  the  same  time  from 
stations  225  miles  apart.  One  takes  |  minutes  longer  than  the 
other  to  run  a  mile,  and  they  meet  in  3  hours.  Find  the 
speed  of  each  train  ? 

42.  A  teamster  having  12  miles  to  drive  increased  his  speed 
one  mile  an  hour  after  he  had  traveled  two  miles.  He  thus 
finished  the  distance  in  half  an  hour  less  time  than  it  would 
have  taken  had  he  not  increased  his  speed.  How  long  did  it 
take  to  drive  the  12  miles  ?     7^0 

Suggestion. — Let  his  first  rate  be  x  miles  an  hour. 

43.  A  merchant  bought  a  certain  number  of  mirrors  for  1 36, 
and,  after  breaking  one,  sold  the  rest  for  50  cents  apiece 
more  than  they  cost  him,  thus  making  $2.50  by  the  transaction. 
How  many  did  he  buy  ? 

.  44.  The  rate  at  which  a  man  can  row  in  still  water  is  twice 
the  rate  of  the  current  of  a  river.  He  rows  6  miles  down  the 
river  and  back  in  4  times  as  many  hours  as  he  could  row  miles 
per  hour  in  still  water.  Find  the  rate  of  the  current  and  the 
rate  of  his  rowing  in  still  water. 

45.  The  number  of  square  inches  in  the  area  of  a  square  ex-  ry 
ceeds  the  number  of  inches  in  its  perimeter  by  32.     What  is 
its  area  ? 

46.  The  side  of  a  square  is  the  same  as  the  diameter  of  a 
circle.  The  area  of  the  square  exceeds  that  of  the  circle  by  ap- 
proximately 3.4336  square  inches.  Find  the  diameter  of  the 
circle. 

47.  The  base  of  a  triangle  exceeds  its  altitude  bj^  2  inches, 
and  its  area  is  112  square  inches.     Find  its  altitude. 

48.  An  engraving  whose  length  was  twice  its  width  was  so 


i- 


U 


252  ALGEBRA 

mounted  on  Bristol  board  as  to  have  a  margin  3  inches  wide, 
and  equal  in  area  to  the  engraving,  lacking  36  square  inches. 
Find  the  width  of  the  engraving. 

49.  A  certain  number  consists  of  two  digits  whose  sum  is 
12 ;  and  the  product  of  the  two  digits  plus  16  is  equal  to  the 
number  expressed  by  the  digits  in  reverse  order.  What  is  the 
number  ? 

50.  The  glass  of  a  mirror  in  the  shape  of  a  rectangle  whose 
length  Avas  twice  its  breadth  cost  $1.25  a  square  foot.  The 
frame,  measured  on  the  inside,  cost  10.75  a  linear  foot.  If  the 
glass  cost  $22  more  than  the  frame,  what  were  the  dimensions 
of  the  glass  ? 

61.  A  certain  farm  is  a  rectangle,  whose  length  is  twice  its 
breadth;  If  it  were  20  rods  longer  and  24  rods  wider,  its 
area  would  be  doubled.  How  many  acres  does  the  farm  con- 
tain ? 

52.  A  square  lot  has  a  gravel- walk  around  it.  The  side 
of  the  lot  lacks  one  yard  of  being  six  times  the  breadth 
of  the  gravel-walk,  and  the  number  of  square  yards  in  the  walk 
exceeds  the  number  of  yards  in  the  perimeter  of  the  lot  by 
340.     Find  the  area  of  the  lot  and  width  of  the  walk. 


CHAPTER  XVI. 

HIGHER  EQUATIONS.    EQUATIONS  INVOLVING 
SURDS.— ONE  UNKNOWN  NUMBER. 

1 70.  It  is  not  within  the  limits  of  this  work  to  consider 
general  equations  of  degree  higher  than  the  second^  but  some 
special  equations  of  higher  degree  may  he  thrown  into  a 
quadratic  form^  i.  e.,  we  may  consider  tlie  unknown  as  being  a 
power  of  some  single  unknown,  or  as  an  expression  involving 
an  unknown. 

Thus,  x^  +  3a?^  +  8=0  is  a  quadratic  in  iir^,  for  if  we  let  J(^=y,  an 
unknown,  we  have  y'^  +  Sy +  8=0,  a  quadratic  in  y. 

{x'  +  3x—2y  +  2{x^  +  3x—2)  —  3=0  is  a  quadratic  in  x'  +  3x—2. 
Letting  y=x'^  +  2x—2,  we  have  y^  +  2y—o=0,  a  quadratic  in  y. 

Such  equations  may  be  solved  as  quadratics. 

Example  1.     Solve  x^-13x'2  + 36=0. 

This  equation  has  the  form  of  a  quadratic  in  x^.  Solving 
for  x^ 


Now  solving 

Solving 

Hence, 

Example  2.    Solve 


_13±T    169- 

Ml 

x-             2 

13±5 

-       2 

=  9  or  4. 

x'^d,               * 

x=±3. 

x^=4, 

x=±2. 

x=S,  -3,  2,  or 

-2. 

x^          x  +  1     , 
X+1+"    x^  -^ 

253 

254  ALGEBRA 

Here    the    second    fraction    is    the    first    fraction    inverted. 
Putting  y  for  -— -^,  the  equation  becomes 

Multiplying  hj  y,         y''—3y  +  2=0. 

Factoring,  (y—2){y-l)=0. 

Hence,  y=2  or  1. 

Therefore  the  given  equation  is  equivalent  to  the  equations 

-— -7=2  and  — r^=l. 
x+1  x+1 

x^ 
Solving  ^+1""^'  ^=^  ±  y^- 

Solving  ^^=1,   .=1^. 

Example  3.    Solve  2x^  +  2x  +  l      ^^ 


x^  +  x' 


This  may  be  written  2{x'^  +  x)  +  l=   .^       . 

X  -\-  X 

Multiplying  by  x"^  +  x,  2{x^  -{-xY-\-{x^  +  x)  —  \Q=Q. 

This  is  a  quadratic  in  x^  +  x.     Solving  for  x^-\-x, 


x^  +  ^.=  -l±l/l  +  80 


4 

=2  or  -f. 

Hence,  the  given  equation  is  equivalent  to  the  two  equations 


x^  +  x=2    and  x^-{-i 


Solving  x^  +  x=2,  x=l  or  —2. 

Solving  x'^  +  x=—^,x= 2~^ * 

Note. — In  every  equation  which  we  have  solved  in  this  book,  the 
number  of  roots,  or  solutions,  is  equal  to  the  degree  of  the  equation.  In 
the  Theory  of  Equations  it  is  proved  that  this  is  true  in  general.  A 
linear  equation  has  one  root,  a  quadratic  tico  roots,  a  cubic  three  roots, 
etc.  The  student  should  see  that  in  every  equation  he  gets  a  number 
of  roots  equal  to  the  degree  of  the  equation. 


HIGHER  EQUATIONS.     EQUATIONS  INVOLVING  SURDS    255 

171.  Equations  of  higher  degree  than  the  second  can  some- 
times be  solved  by  factoring. 

Example  1.     Solve  xHl=0.  (1) 

Factoring,  {x+l){x^—x+l)=0. 

Hence  (1)  is  equivalent  to       a:? +  1=0  and  x^—x-\-l=0. 

Solving  a? +  1=0,  x=  —  l. 

Solving  x^-x  +  l=0,  x=^±^]/'^. 

Example  2.     Solve  x^-U=0.  (1) 

Factoring,  (x  +  2){x- 2) {x^  +  2x+4)(x'-2x-{-4)=0. 
Hence,  (1)  is  equivalent  to 

i»+2=0,  a?— 2=0,  x^  +  2x  +  4:=0,  and  x^—2x  +  4=0. 
Solving  each  of  these,  x=—2,  2,  — 1  ±  j/— 3,  or  1  ±  ]/— 3. 

Examples.     Solve  ir'  +  5a?=6a?^— 12.  (1) 

Adding  -6.^2  +  12,  ar'-6icH5if  +  12=0. 
Factoring  by  the  remainder  theorem, 

{x-3){x  +  l)(x-4)=0. 
Hence,  (1)  is  equivalent  to  a?— 3=0,  a7+l=0  and  a?— 4=0. 
The  solutions  of  these  equations  are  3,  —1,  and  4,  respectively. 


Solve  the  following : 

1.  i«*-16-0. 

2.  x*-2x'  =  l^, 

3.  2a;*-5a;^-12  =  0. 

4.  Q^  +  2x'=x  +  2, 

5.  ar*-a;'-4a;  +  4  =  0. 

6.  a;^-2a;^-5a;  +  6  =  0. 


11.    1--X 

.X 


EXERCISE 

!  79. 

7. 

1    .x'^-l 
x^'^     6     " 

=  1. 

8. 

0.^  +  1=3. 

9. 

(a;^-iy  +  24= 

=  ll(a;''- 

-!)• 

10. 

{x^^tcy- 

-2(a;^  +  2a;): 

-3. 

-11( 

n     ^ 

X  , 

1+30  =  0. 

256 

ALGEBRA 

1^  l4s  +  =^'-?'=B^ 

17.  x'  =  \. 

18.  a;*-81  =  0. 

19.  i««-l  =  0. 

j4  2x  +  l       «'    _„ 

20.  .T^  +  1  =  0. 

^*'      x'      '  2a!  +  l     ^• 

15.  .^  +  1-^4,. 
16..'  +  .     1-/^^. 

21.  £c^-256-0. 

22.  i««-16  =  0. 

23.  ;«''  +  l  =  0. 

24.  (Saj'^  +  Ga;)^ 

=  1 

-(.?.''^  +  3a^  +  2). 

25.  {x^-x  +  4.) 

^  + 

{x'-x)  =  2. 

26.   (.  +  ?)'  + 

(' 

4)=- 

27.  a3H-a^  +  -4- 

X 

1 

=  4. 

Suggestion,   x' 

«-|.=("S'- 

28.  a;*  +  3ic^-2ic^-3a;+l  =  0. 

Suggestion.  Divide  by  x^  and  arrange  as  in  Ex.  27. 

29.  x'-%x^^^^x^-Zx-\-l  =  ^. 

172.  Equations  which  involve  surds.  To  solve  equations 
which  are  *>ra^/o?ia^  with  respect  to  the  unknown  number,  it  is 
necessary  to  free  the  unknown  number  from  the  radical  sign. 
To  do  this  both  members  of  the  equation  must  be  raised  to  the 
same  power.     (Ax.  5.) 

The  following  rule  may  be  used  in  general : 

(1).  Transform  the  equation  into  an  equivalent  one  in  v^hich  one 
member  consists  of  a  single  surd.  This  is  called  isolating  the  surd. 

(2).  Raise  both  members  to  such  a  power  as  will  free  this  iso- 
lated surd  of  the  radical  sign. 

(3).  If  surds  still  remain.,  repeat  the  operation. 


HIGHER  EQUATIONS.— EQUATIONS  INVOLVING  SURDS    257 
173.  Introductionof  new  solutions. 

In  general^  icheri  both  members  of  an  equation  are  raised  to 
a  higher  power ^  new  solutions  are  introduced. 

For,  let  any  equation  be  represented  by 

A  =  B. 

Then,  squaring  both  members, 

A'  =  B\  or  A'-B'  =  0. 

Factoring,  {A-B){A  + B)=0, 

This  is  equivalent  to  A  —  B  =  ^  and  A^^B  =  0. 

Of  these  two  equations  A—B=0  alone  is  equivalent  to  the 
given  equation.  Hence,  all  of  the  solutions  of  /l+B  =  0  are 
introduced  by  squaring. 

It  is  necessary,  therefore,  in  solving  equations  which  involve 
surds,  to  examiyie  all  solutions  and  discard  those  which  do  not 
satisfy  the  equation  in  its  original  foi'm. 

Example  1.  Solve          |/ 2^—3  =  5. 

Squaring,  2x—S=25. 

Whence  2x=28. 

And  ir=14. 

The  solution  14  satisfies  the  original  equation,  and  therefore 
was  not  introduced. 


Example  2.     Solve  ^  +  y'x-\-7=x. 

Isolating  the  surd,  \/x  +  7 = a? — 5. 

Squaring,  x+7=x'^  — 10a:j  +  25. 

Or  x'—llx-^l^^O. 

Factoring,  {x—^){x—2)=0. 

Therefore,  ic=9  or  2. 

Now  9  satisfies  the  given  equation,  but  2  does  not.     Solution  2 
was  introduced  by  squaring. 
17 


258  ALGEBRA 


Example  3 .     Solv^e  y  2ic  -f  8  +  2  +  2  ]/a?  +  5 = 0 . 

Isolating  second  surd,  i/2x  +  8-{-2=  —  2yx-{-5. 

Squaring,  2x  +  8  +  4:  +  4^/2x  +  S=4x  +  20. 

Therefore,  2\/2x-\-8=x  +  4.. 

Squaring  again,  8aj+32=ic''  + 8^7+16. 

Or,  0^2^16. 

Whence,  07=4  or  —4. 

Now  neither  4  nor  —4  will  satisfy  the  given  equation.  Both 
solutions  were  introduced  by  squaring.  Hence  the  equation 
has  no  solutions  ;  it  is  an  impossible  equation. 


Example  4.     Solve  3ic'— 4ic+]/3a;'— 4a;— 6  =  18. 

Equations  such  as  this,  where  the  unknown  enters  alike  in  the 
part  free  from  radicals  and  under  the  radical,  may  be  solved  as  a 
quadratic. 


By  adding  —6,  we  have  3a?^— 4a?— 6  +  ]/3a?''— 4ic— 6=12,  a  qua- 
dratic in  y2,x^  —  4:X—%. 


l±l/l  +  48 


Hence,    ]/3x^— 4a7— 6  = -^ =  3  or —4. 


The  solutions  of  |/3x^— 4£C— 6=3  are  3  and  —  |. 

And  |/3ic^— 4ic— 6=— 4  will  be  found  to  be  an  impossible  equa- 
tion. 

Hence  the  required  solutions  are  3  and  —  |. 

Example  5.     Show  that  1  has  three  cube  roots,  and  find  their 
values. 

We  are  to  find  what  numbers  cubed  will  equal  one.     If  we  let 
X  stand  for  the  cube  root  of  1,  then 

a^=l. 
Adding  -1,  a^-l=0. 

Factoring,       {x—l){x^  +  x-\-l)=0. 
Therefore,  a;— 1=0,  3(^  +  x+l=0. 

The  solution  of  a?— 1=0  is  1. 
The  solutions  of  a;Ha?+l=0  are  — |  +  ^i/— 3  and  — |— ij/— 3. 


HIGHER  EQUATIONS.— EQUATIONS  INVOLVING  SURDS    259 

Hence,  there  are  three  cube  roots  of  1,  one  a  real  number  and 
the  other  two  complex  expressions  ;  viz.  : 

1,  — 2  +  ^1/— 3,  —\     ^]/  — 3. 

EXERCISE   80. 

Solve  the  following : 


1.  i/aj+l==2.  4.  4i/ic  +  5  =  3]/3i«4-4. 


2.  8-|/a;-l  =  6.  5.  5i/ic  +  2-3i/4ic-3  =  0. 


3.  6  =  10-2i/5a;-l.  6.  iri3  +  «=2. 

7.  f/3^M=T6-if2^M=3^T20  =  0. 


8.  i/15  +  3a;  +  2  =  i/23-a-.  17.  yx  +  2-l  =  \/x-^. 

9.  2i/9T^=3i/^T24-10.       18.  ^=-f2x-^. 

10.   Vx'  +  ^x-r^=x  +  l.  1^'  V^=^=J-^' 

20.  i/aj+i/32  +  iK  =  16. 


11.  cc— 3  =  i/2i«"'— ic  +  lO  — 5. 


12.  i/5iK  +  4-i /12ic  +  21. 


21.  i/£c  +  4=-2  +  |/a;. 


13.  \/x=yx  +  'lb  —  l. 


22.  \/x+Q  =  vl2  +  x, 


23.  ^/x-S2  =  lQ-i/x. 


14.  i/«.-12  =  i/-«-2.  24    1/^^3-1/^+12::=-^ 

15.  1/^-7  +  1/^+7  =  0.  25.  a!+i/^T5  =  5. 

16.  2j/^+2  =  i/^.  26.  1/2^^  +  1/2^+9  =  8. 

Ql 

27.  i/3a;  +  i/3a;  +  13 


28.  i/aj-i/a;-3-^- 


l/3a;  +  13 

2 


29.  i/a;^-3a;  +  5-i/a;^-5ic-2  =  l. 


30.  ■~T~  =  \/x+i/x-l,  31.  i/cc  +  i/ic-6  = 


l/aj     •^-•--     -  -"•   ^^•^^'/^-"-l/-^:=r6- 


260  ALGEBRA 

32.  x'-Sx  + 1  +  2yx'-^x=4.     39.  y^^:f^j^^y^:f^  =  Q^ 


40   ^—= —  =2. 

34.  i/i«  +  2-5--i/ic-3.  '  yl  +  x-yl-x 

Vx-\_    X  41.  :^— =__!—._     8    _^ 
00.  ■  ,— I  -,  — riTT'  1  — |/£c     1  +  1 /cc     1— .« 


y'x  +  1     iG  —  1 


y'x+\  —  \/x—\ 


36.  ■\/x+yl+x=^/Yjf^:  '  yx  +  l  +  y'x—1 


Vl  +  x 


2  43.  ^/x+Vl  —  i/x  +  x^^l. 

37.  i/£c— 1   i«— 


l/a;— 8  44.  i/l—x+i/2—x  =  yl—4x 

38-  |/^^^^a^-r*/^H^"3V.  45.  i/a^^  +  3ic  +  l-l-2a;^-6a;. 


47. 


46.   yx  +  2  +  i/x—S  =  y'2x+ll. 
20  _         ._  12 


=  -l/«.=  v/15+a..       48.  -^^==.=  ,/^^  +  s  +  x. 


1/15 

49.  i/xTlO  +  i/x^^=yQx. 
Solve  for  a? : 


60.  i/ic  +  a'^— i/ic— a^  =  l/26.       52.  y^x+ya—x=y'a-\-b. 

_        2a3  

61.  i/x-\-\/x  +  a=—=^-  53.  i/aj— a  =  ^>— i/^. 

54.  Find  the  four  fourth  roots  of  16. 
66.  Fhid  the  six  sixth  roots  of  1. 


CHAPTER  XVII. 

SYSTEMS  INVOLVING  QUADRATIC   AND   HIGHER 
EQUATIONS. 

174.  As  in  the  case  of  linear  equations,  so  in  general,  solv- 
ing a  system  of  quadratic  equations  requires,  first,  the  elimina- 
tion ofallhutone  of  the  unknown  numbers  ;  and  second,  so?y*?i^ 
the  resulting  equation  for  that  unknown  number. 

When  a  system  involves  quadratic  or  higher  equations,  the 
method  of  elimination  depends  upon  the  forms  of  the  equa- 
tions. 

175.  One  equation  linear  and  one  quadratic. 

When  one  equation  is  linear  and  one  is  quadratic,  either  of 
the  unknown  numbers  may  be  eliminated  by  substitution ; 
and  the  elimination  leads  to  a  quadratic  equation.  Since  a 
quadratic  equation  in  one  unknown  number  can  alicays  be 
solved,  then  a  system  consisting  of  a  linear  and  a  quadratic  can 
always  be  solved. 

Example  1.    Solve  |  ^.-^I?7.  g| 

Solving  (1)  for  a?,  ir=6  +  22/.  (3) 

Substituting  this  value  of  x  in  (2), 

42/' +  241/ +  36  4-2/'= 17.  (4) 

Solving  (4) ,  2/  =  - 1  or  - »/.  ^  v 

Substituting  these  values  of  y  in  (3),  \X.""*v 

when  2/=— 1»  ic=6— 2=4;  ^ 

when  2/=-¥,  a^=6-^8-=-f-  ^^  ^ 

Hence,  there  are  two  solutions  :  v-^  "*\ 

a?=4,  2/=-l;  ^=-1,  2/=--/. 
Let  the  student  check  the  result. 

261 


262 


ALGEBRA 


EXERCISE  81. 


Solve: 


4. 


5. 


7. 


10. 


11. 


<x  +  y  =  b, 


2x  =  12, 

Sx  +  Q=0. 


Sy-2x  = 

Xy^—x^ — 

(x'  +  y'  =  74, 
\Sx-2y  =  h 

\ 


x-y  =  l, 
xy  =  2. 

\x-y  +  S  =  0. 

2x  +  2y  =  bxy, 
2x  +  2y  =  b. 


{ 

(2a;'^  +  3?/  =  ll  +  4a;y, 

\x- 

{ 


x  +  y  =  7, 
x'^—xy  +  y^  =  Sl. 

x-^y  =  4, 
X     y 


12. 
13. 
14. 
15. 

16. 

17. 

18. 

19. 
20. 

21. 


(  x  +  y  =  ^. 
ix-S=y, 
I  x'^  Id+y'^Sxy. 

j8ic  +  12-4y, 
(3a3^  +  2/-48=y. 
J  2x—b=y^ 
I  x  +  Sy  =  2xy. 

x  +  y  =  2, 

y     X 

x'  +  xy  +  y'  =  97, 

1-1  =  1^. 

X        X 

3 

U        t  3       ' 

3^-2^^  +  12  =  0. 
(  w^  =  3n  +  l, 

ri__^+^^=2i 


J52» 


^^B=B' 


176.  Two  quadratic  equations  of  the  form  ax'^  +  by^=c. 

In  general,  solving  a  system  which  contains  tico  quadratic 
equations  involves  the  process  of  solving  an  equation  of  a 
higher  degree  than  the  second,  in  one  unknown  number. 


Thus,  consider  the  system 


(1) 
(2) 


QUADRATIC  AND  HIGHER  EQUATIONS  263 

Solving  (1)  for  a?,  x=7—2y^. 

Substituting  this  value  of  x  in  (2), 

49-28i/=^  +  4t/*  +  32/=6. 
This  equation  is  of  the  fourth  degree  in  y. 

There  are,  however,  some  such  systems  which  lead  to  the 
solution  of  quadratic  equations.  There  are  systems  in  which 
one  unknown  can  be  eliminated  by  addition  or  subtraction; 
and  the  elimination  leads  to  a  quadratic  equation. 

EXAMPLE  1.     Solve  I  fXY^=f7:  gj 

Multiplying  (1)  by  2,  2a;'  +  4i/''-44.                                   (3) 

Subtracting  (2)  from  (3),  ^y'=27, 

whence,  y=±S. 

Substituting  3  for  y  in  (1),  x' +  18=22, 

whence  x=:t2. 

Substituting  -3  for  y  in  (1),  x' +  18=22, 

whence  x=±2. 

Hence  there  are  four  solutions  : 

x=2,  2/— 3;  x=—2,  y=^\  x=2,  y=  —  S',  x=—2,  y=—S. 

Note. — In  all  of  tlie  systems  of  equations  which  we  have  solved,  the 
number  of  solutions  is  equal  to  the  product  of  the  degrees  of  the  two 
equations.  This  is  true  in  general.  If  one  equation  is  of  the  mth 
degree  and  the  other  of  the  ?ith  degree,  the  number  of  solutions  will 
be  mn.  Exceptions  to  this  rule  in  certain  peculiar  systems  will  be  noted 
later. 


EXERCISE   82. 


Solve : 


„     I  35^  +  4?/'^  =  40,  .     r4a;^  +  3y^^43, 

3     r  0^^-21/^  =  17,  g     r9a;^  +  /=29, 

^'    |2a3=^  +  y^  =  54.  ^-    |  27aj-^-2y^  +  38  =  0. 


264  ALGEBRA 


7. 


la;       16x'  ■^^'    ■|a;^  +  ll  =  3yl 


177.  Two  quadratic  equations  ;  one  of  them  homogeneous. 
A  homogeneous  equation  is  one  whose  terms  are  all  of  the 
same  degree. 

Thus,  2x'^—xy-{-y'^=0  is  homogeneous.  Observe  that  a  homo- 
geneous equation  has  no  term  free  from  unknown  numbers. 

The  process  of  solving  a  system  of  two  quadratic  equations, 
one  of  which  is  homogeneous,  may  be  reduced  to  the  process 
of  §  175. 

EXAMPLE  1.     Solve  I  ^f^+^^;=;^  g) 

Factoring  (1),                    y{2x  +  y)=i).  (3) 
Equation  (3)  is  equivalent  to  the  equations 

^-0,  (4) 

and                                              2x  +  y=0.  (5) 

Now,  since  all  solutions  of  (4)  and  (5)  are  solutions  of  (1),  and 

since  all  solutions  of  (l)are  solutions  of  (4)  or  (5),  the  given  system 

i3  equivalent  to  the  two  systems, 

a\                   2/=0,  (4) 

^(  2^  +  2/-0,  .  (5) 

^\  x''-xy  +  y'=2S.  (2) 

The  solutions  of  systems  A  and  B  are  obtained  by  the  method 
of  §  175. 

The  solutions  of  A  are  £c=]/28,  y—0\  x=  — ]/28,  2/=0. 

The  solutions  of  B  are  x=2,  y=—4;  x=—2,  i/=4. 

These  four  solutions  are  the  required  solutions  of  the  given 
system. 

See  that  these  solutions  satisfy  both  of  the  given  equations. 


or 


J  5x'  +  4xy-y'=0, 

{  x^  +  x  +  y=5. 

(1) 
(2) 

?*f— ■ 

+  4( -)— 1=0,  a  quadratic  in  -. 

(3) 

-=l  or  -1. 

(4) 

QUADRATIC  AND  HIGHER  EQUATIONS  265 

ExAitfPLE  2.     Solve 
Dividing  (1)  by  y"^ 

if) 

Solving  (3)  for  |,  ^ 

Multiplying  (4)  by  51/ and  2/ respectively,  5x—y=0  or  x-\-  y=0.  (5) 
This  gives  the  systems, 

.  (  5x-y=0, 

^j  x'  +  x  +  y=5; 

^\  x'  +  x  +  y=5. 
From  A,    x=  —  3  +  ylT,    y=  —  la  +  5yU\ 

x=-3-i    Ti;   2/=-15-5]/147 
From  5,    x=y%   y=:  —  i/5;   x=  —  y^,    y=y''^. 

The  method  used  in  Example  1  might  have  been  used  also 
in  Example  2.  It  is  evident,  however,  that  the  method  of 
Example  2  is  the  more  general.  It  may  be  used  in  case  of  any 
homogeneous  equation,  while  the  method  of  Example  1  may  be 
used  only  in  case  the  first  immber  of  the  homogeneous  equation 
can  be  factored. 

178.  All  unknown  terms  of  the  second  degree  in  each  equation. 

Systems  in  which  all  unknown  terms  are  of  the  second 
degree  in  each  equation  may  be  solved  by  forming  an  equiva- 
lent system  of  the  form  discussed  in  §  177.  This  involves 
combining  the  tAVo  given  equations  so  as  to  form  a  homogeneous 
equation,  which  may  then  be  used  with  one  of  the  old  equa- 
tions to  form  a  new  system. 

EXAMPLF  1      Solve      i  ^'-7a?2/-92/''^=9,  (1) 


2Q^  ALGEBRA 

Evidently,  we  may  get  a  homogeneous  equation  by  eliminating 
the  term  free  from  unknowns. 

Multiplying  (1)  by  5,  5x^-S5xy-4:5y''=45.  (3) 

Multiplying  (2)  by  9,  9x''  +  4:5xy-{-9Qy^=45.  (4) 

Subtracting  (3)  from  (4),    4:X'-\-80xy+U4y''=0, 

or  x''  +  20xy  +  3Qy^=0,  (5) 

Forming  a  new  system  of  (5)  and  (2), 

20xy  +  3Qy^=0,  (5) 

5xy  +  lly''=o.  (2) 

We  thus  have  the  kind  of  system  discussed  in  §  177. 
Factoring  (5),        {x  +  2y)ix+18y)=0, 
whence,  x  +  2y=0^  x+18y=0. 

With  these  and  equation  (2)  we  may  form  two  systems, 
„(  xi-2y=0, 
^}  x'  +  5xy  +  lly'=5; 

5xy  +  ny^=5. 
From  J5,  a?=2,  y=  —  l;  x=—2,  y=l. 


J  x'  +  ^ 
I  x'  + 


^ixf+ 


From  O,    a?=y,  2/=-!;  -^^^-V,  2/=t- 


Solve  : 
1. 


EXERCISE  83. 


ix''—^xy+2y'  =  0,  ^  ( x^  +  xy—2y'  +  U  =  0, 

o     (  2£c^  +  3a!y  =  0,  «  (a;=^  +  «y  +  2/'  =  39, 

|a;'-a;y  +  2/^  =  19.  °'  \2x'-^Sxy-^y'  =  m. 

„     (3xy  +  5/  =  0,  Q  («^^  +  2a^y  =  24, 

^'    X^x'  +  4xy  =  l^.  ^'  |2a;y  +  4/  =  120. 

-     ix'  +  xy  =  21,  .Q  (i«^  +  c«y  +  2y^:=74, 

•    (2a5y-2/^  =  8.  •""•  (2iK^  +  2icy  +  y^  =  73. 

\x^  +  xy  +  2y^  =  4:4:.  '  \x^—xy  +  y^  =  21. 

g     (a;^-3a^i/  +  2y^=3,  ^o  ( a^^-a^y-12y^-8, 

'*•    l2a;^  +  y^  =  6.  '^'  \x'  +  xy-10y'  =  20. 


QUADRATIC  AND  HIGHER  EQUATIONS  267 

179.  Symmetrical  equations,— one  quadratic,  the  other  linear 
or  quadratic. 

A  symmetrical  equation  is  one  which  is  unchanged  when  the 
unknown  numbers  are  interchanged. 

Thus,  £c^  + 2/^=5  and  xy=2  are  symmetrical. 

Some  systems  of  two  symmetrical  equations  can  be  solved 
by  the  methods  discussed  in  the  preceding  sections.  Another 
good  method  of  solving  a  system  of  two  symmetrical  equations 
in  X  and  y  is  first  to  find  the  values  of  ic  +  y  and  oix—y. 

Example  1.     Solve         {"^^Zfi  gj 

Multiplying  (2)  by  2,  2xy=24:.                        (3) 

Adding  (3)  to  (1),  x''  +  2x7j  +  if=64. 

Extracting  the  square  root,  x+y=8  or  —8.              (4) 

Subtracting  (3)  from  (1),  x'-2xy  +  y^=16. 

Extracting  the  square  root,  x—y=4:0r  —4.              (5) 

The  old  system,  then,  is  equivalent  to  the  four  systems 

i  x  +  y=8,  {  x  +  y=-8,      ^i  x+y=8,  (  x  +  y=-8, 

^\x-y=4.;      ^\x-y=4.-         '-\x-y=-4.;      ^\  x-y=-4. 

The  solution  of  A  is  x=Q^  y=2. 
The  solution  of  B  is  x=—2,  y=  —  6. 
The  solution  of  O  is  x=2^  2/= 6. 
The  solution  of  D  is  a?=  — 6,  y=—2. 

When,  except  for  the  sign,  the  equations  are  symmetrical, 
the  system  may  be  solved  by  a  similar  method. 

EXAMPLE  2.    Solve         l^-^rJi^,  g 

Squaring  (1),  x^—2xy  +  y^=4.  (3) 

Subtracting  (3)  from  (2),  2xy =70.  (4) 

Adding  (4)  and  (2),  x^  +  2xy  +  y''=lU.  (5) 

Extracting  the  square  root  of  (5), 

x+y=12  or  —12.  (6) 


268  ALGEBRA 

The  given  system  is  equivalent  to  the  two  systems, 

^  (  x  +  y=12;  ^  I  x  +  y=-12. 

The  solution  of  A  is  x= 7,  y=5. 
The  solution  of  5  is  x=  — 5,  y=—7. 

ExAMPLF  S      Solve         \x'  +  y'-x-y=22,  (1) 

J1.XAMPLE  6.     bolve  \^^y  ^  ^y  ^_i^  (2) 

Multiplying  (2)  by  2,    *      2x  +  2y  +  2xy=-2.  (3) 

Adding  (3)  to  (1),       x^  +  2xy i-y""  +  x  +  y=20, 
or  {x+yy  +  {x  +  y)-20=0.  (4) 

Now  (4)  is  a  quadratic  in  x  +  y. 

Factoring  (4),  {x-\-y  +  5){x  +  y—4:)=0. 

This  is  satisfied  when  a^  + 2/ +  5=0;  whence  x-]-y=~5;  (5) 

or  x+?/— 4  =  0;  whence  0?  +  ?/= 4.  (6) 

Hence,  the  given  system  is  equivalent  to  the  two  systems, 

.  {x  +  y  +  xy=-l,  (2)  j^  ^  x  +  y  +  xy=-l,  (2) 

^\x+y=-5;  (5)  ^lx  +  y=4.  (6) 

The  solutions  of  A  are  ir=  — 4,  ^=  —  1;  x=  — 1,  ?/=— 4. 

The  solutions  of  B  are  x=5,  y—  —  l-^  x=  — 1,  y=^. 


EXERCISE  84. 


Solve : 


.  {x^y  =  l,  „     ('a^-?/  =  4,  ;^     {x-y  =  ^, 

^'  |£cM-y'=5.  ^'    |£c^+/-40.  ^'    |icy  +  15  =  0. 

'*•  |£c^  +  y-29.  |£c^  +  /  =  26.  ''•    Ix^-y^ll. 

„  (  x'  +  xy  +  y'  =  4,  .«     (  a^^-a^y  +  y'^eS, 

'•  ji«^-a;y  +  y^  =  2.  ■""•    |i«-2/--3. 

«  (x  +  xy  +  y  =  29,  ..     |  a;^  +  a;y  +  y^  =  39, 

°*  (£c^+£cy  +  2/'=61.  |ic  +  y  +  2-0. 

Q  Ca;?/  +  7-0,  .„     f  a^'-a^y +2/'  =  21, 

^'  {x'  +  f^bO,  ■''*'    |a;+y-9. 


QUADRATIC  AND  HIGHER  EQUATIONS  269 

180.  Systems  involving  equations  of  a  higher  degree.  We   are 

able  to  solve  some  special  systems  involving  equations  above 
the  second  degree.  If  the  equations  are  symmetrical,  or  sym- 
metrical except  for  the  sign,  the  systems  may  often  be  solved 
as  shown  in  the  following  examples : 

Example  1.     Solve  |  f  +  ^^=^;  g 

Dividing  (1)  by  (2),  x^-xy^y'^Z.  (3) 

Squaring  (2),  x^^-'^xy-\-y'='^.  (4) 

Subtracting  (3)  from  (4),    ,.  3a7z/=6, 

or  xy—2.  (5) 

Subtracting  (5)  from  (3),    x^-2xy^y''=\.  (6) 

Extracting  the  square  root,  x—y=\^  or  —1.  (7) 

HQnce,  the  given  system  is  equivalent  to  the  two  systems, 

.\a^  +  2/=3,     (2)  (^  +  2/=3,        (2) 

^\x-y=\',    (7)  ^\x-y=.-\.    (7) 

The  solution  of  A  is  £c=2,  y=\. 
The  solution  of  J5  is  x'=l,  y=2. 

Note.  This  system  has  only  two  solutions,  whereas  we  expected 
three.  Systems  of  this  sort,  in  which  the  number  of  solutions  is  lesa 
than  the  product  of  the  degrees  of  the  equations,  are  called  defective 
systems. 

EXAMPr..2.     solve         |  f  ^.^'z^"'  g 

Leta?=a  +  &,  and  i/=a— 6. 

Substituting  in  (2),  a  +  &  +  a— 6=3, 

whence,  a=|.  (3) 

Substituting  in  (1),     (a  +  6)*  +  (tt— 6)*=641, 
or  2a*  +  12a262  +  26*=641.  (4) 

From  (3)  and  (4),  V +276'  +  2&*=641, 

or  166^  +  21662-5047=0.  (5) 

From  (5),  6=^=-V-  or  — 1|^, 

whence,  6=  ± |  or  ±  1^^-103,        (6) 


270  ALGEBRA 

Using  the  value  of  a  and  each  of  the  four  values  of  6,  we  have 
x=a-^b=5,  -2,  ^  +  |]/-10a,  or|-^i/-103. 


y=a-b=-2,  5,  |-|v  -1^3,  or|  +  ii/-103. 
Hence,  the  four  solutions  are 


x=5,  y=-2;  x=-2,  i/=5;  a?=|  +  ^|/-103,  y=§-^\/  —  10S-  and 


EXEKCISE  85. 

Solve  : 
.      (a3='  +  y-35,  y     j;^*-y*  =  544, 

=  27,  g     (^^  +  a^y  +  2/*  = 

3.  |£C^-a;y  +  y  =  9. 

«^y  +  /  =  28. 


2     U^^-/^27,  g     f^^  +  a^y +  2/^  =  243, 

3. 


|i«-y  =  7.  •  (.73^  +  a;y  +  /  = 

|a^  +  y  =  3.  ■^"'  \x  +  y=b. 

f.r^+y  =  504,  .,  (a.^  +  y^  =  82, 

°'    \x'-xi/  +  f  =  M.  ^^'  \x  +  'i/  =  4:. 

(^^  +  y  =  257,  .o  ix'-i/'  =  19, 


19. 


181.  Special  devices.  In  the  preceding  sections  we  have 
discussed  a  few  types  of  systems  involving  quadratic  and  higher 
equations.  The  method  of  solving  such  systems  is  determined 
by  the  fonn  of  the  equations  in  the  systems.  A  careful  study 
of  each  system  will  often  reveal  some  special  device  for  solving 
it.  Some  illustrations  are  here  given  showing  special  devices 
for  solving  types  of  systems  unlike  those  heretofore  dis- 
cussed. 

EXAMPLE  1.     Solve     I  ^^*+,t^;=^,»'  g 


QUADRATIC  AND  HIGHER  EQUATIONS  ^71 

Factoring  (1),  (ar'2/'  +  5)(^V-4)=0, 

whence,  x^y^  +  5=0,  (3) 

or  x'y'-4.=0.  (4) 

Solving  (3),  xy=  ±  i/=^.     Solving  (4),  xy=:  ±  12. 

Hence,  the  old  system  is  equivalent  to  the  four  systems, 

\xy=l/-5;  lxy=-V  -5;        \  xy=2.,  i  xy=-2. 

These  systems  will  give  in  all  eight  solutions. 

x-^y=W'  ^^^ 

Example  2.    Solve      \^    ^    "^.^ 

Let  -=a,   -^b. 

^  X       '  y 

Then  (1)  becomes                           a  +  b=^,  (3) 

and     (2)  becomes                             a^  +  b^=^,  (4) 

Squaring  (3),                         a'  +  2ab  +  b''=ll.  (5) 

Subtracting  (4)  from  (5),                 2ab=l^.  (6) 

Subtracting  (6)  from  (4) ,  a'  -  2ab  +  b^= ^V-  (7) 

Taking  the  square  root  of  (7),      a— 6=^  or  — |. 

Hence,  we  get  the  systems, 

These  systems  give  a=^,  ^=3;  and  a=|,  b=^. 
Hence,  x=2,  2/=3;  and  x=3,  y=2. 

Observe  that  this  could  have  been   considered  a  system  in 

-  and  -,  and  we  should  have  had  the  same  solution.     The  sub- 
X  y 

stitution  was  to  save  writing  the  fractions  so  often. 

EXAMPLES.     Solve     \^Zt=Zt^Lny.  g 

Arranging  and  factoring  (1),      (x  +  y){x—y—l)=Oy 

whence,  x  +  y=0,  or  x—y—l=0.  (3) 

Arranging  and  factoring  (2),  (x— 5)(.r— 32/)=0, 

whence,  a?— 5— 0,  or  x— 32/=0. 


272  ALGEBRA 

Hence,  the  given  system  is  equivalent  to  the  four  systems : 

^  |ic-5=0;      -^lx-Sy=0;     ^\x-5=0;  -^  {  x-Sy=0. 

The  solutions  of  these  systems  are 

x=5,  y=  —  5-  x=0,  y=0;  x=5,  y=4:]  and  x=|,  i/=|. 


EXERCISE  86. 

Solve : 


1    /l   ic  +  i/2/=8, 
•   \x'  +  y'  =  70Q. 


{: 


Suggestion.     Let  '\/x=v,  \/y=w. 

x  +  y-^xy  =  9, 

£c'^  +  y^— a?— y=  12. 

Suggestion.     Find  the  values  of  x~\-y. 

„     ri«*  +  »?V  +  2/'  =  481, 
:    \x'-{-xy  +  y'  =  Sl.  ■ 

.X     y 

Suggestion,   het -z=a,  -=b. 

X         y 


C     3  R     f  a^y  +  ir.?/— 6  =  0, 

yx-Vy-^    ,--T— =4,  ^"    1  jc^  +  v'  =  5. 

j  x'^  +  f_b^     ^  |i«  +  l/a;y  +  2/  =  14, 

I    xy        14*  '•    |a;2  +  icy  +  y'=84. 

Suggestion.     Divide  one  by  the  other. 

«     {x'  +  xY  +  y'=9l,        ■  Q     {x'y-x=bO, 

°'    jirM-icy  +  2/^  =  13.  ^'    |icy-aj^  =  2475. 

Suggestion.     Divide  one  by  the  other. 

10.  iT-^^i-'  1 


QUADRATIC  AND  HIGHER  EQUATIONS 

Suggestion.     Let  f^x=a,  f^y=b. 

13 


273 


11.    f  »''  +  y'  +  7i«y  =  171, 


xi/=2(x+:i/). 


12.    |^^+2/^-(^-y)  =  20, 


xy^x-y  =  \. 


"•{ 


£c  +  y  +  2i/^+y  =  24, 
£c— 2/  +  3j/a;— y=10. 


Suggestion.     In  Ex.  14.     Let  ^a7+i/=a,  ^/a?— 2/=6. 


jcV  +  a^/^lSO, 


Suggestion.     In  Ex.  16,  factor  each  equation 


£^2^2^400. 


^^'     (;r-^-2^y  +  /  =  3(^-y). 


182.  Geometric  pictures  or  graphs 
that  the  geometric 
picture,  or  graphs 
of  a  linear  equation 
with  two  unknown 
numbers  is  always 
a  straight  line.  It 
was  also  shown 
that  the  solution 
of  a  system  of  two 
linear  equations  is 
represented  by  the 
point  of  intersec- 
tion of  their  graphs. 

The  graph  of  an 
equation  of  higher 
degree  than  the 
first  in  two  un- 
knowns is  usually 
one  or  more  curved 
lines. 
18 


In    §  134   it   was   shown 


Y' 

E, 

^ 

^^^* 

D 

^ 

y^ 

<v 

/ 

R 

/ 

X 

4 

0 

t 

F 

\ 

d< 

\ 

^ 

V 

v.. 

Y 

/^ 

^ 

^ 

Fig.  7. 


274 


ALGEBRA 


Example  1.     Draw  the  graph  of  y^=4:X-\-l. 

Solving  for  x,  x=^-r — • 

Assigning  values  to  y,  and  finding  the  corresponding  values  of 
Xy  we  get  the  following  solutions : 

(^=-1,    J  x=0,     <^x=i,    \x=2, 
\y=0-       }  y=l-    \y=2;    \y=S', 
x=^,    J  x=0,        {  x=^,        J  x=2,        f  ic=3|,  . 
2/=4;     1  y=-l',    I  y=-2;   ]  y=-3;   \  y 4. 

Locating  the  points  whose  coordinates  are  these  solutions,  we 
get  the  points  A,  B,  C,  A  E,  F,  G,  H,  J,  etc.,  Fig.  7.  If  we  join 
these  points  by  a  smooth  line,  we  get  the  curve  shown  in  figure  7. 

This  curve  is  called  a 
parabola.  It  is  the  kind 
of  path  in  which  astrono- 
mers have  found  that 
most  comets  move. 

Example  2.     Draw  the 
graph  of  4a?^+z/^=16. 
Solving  for  y, 


y=±yU~^x\ 

Assigning  values  to  a?, 
and  finding  the  cor- 
responding values  of  y, 
we  get  the  following  solu- 
tions : 


?/=4: 


X— 

y=yi5 


;    1   2/=-i/l5; 

{x=—l, (  a7=— ^, {  x=l, j  ^=1,      j  x=—l, 
2/=|/l5;     |2/=-t/15;     \  y=i/l2;      {  y=-i/12-    \  y=i/12', 

iX^       1,  J   X=^^  J   X=-2,        J   £C=       2> 
y=-l/12;      {y=V7;      \y=.-y7-     \y\\V'^\ 
J  x=-h  _     j  x=2,      j  x=-2, 

i  y=-v7;    1 2/=o;    1 2/=^;   etc. 


QUADRATIC  AND  HIGHER  EQUATIONS 


2Y5 


Now  proceeding  as  in  Example  1,  we  get  the  curve  shown  in 
Fig.  8. 

This  curve  is  called  an  ellipse.  Astronomers  have  found  that 
all  of  the  planets  move  in  ellipses. 

Example  3.     Draw  the  graph  of  a?H  i/^=9. 

By  proceeding  as  in  Example  2,  we  get  the  circle  shown  11 
Fig.  9. 


Fig.  9. 

183.  Solutions  of  systems.  The  solutions  or  roots  of  a  sys- 
tem are  the  coordinates  of  the  points  where  the  graphs  of  the 
equations  cross^  or  intersect. 

Example  1.     Interpret  the  solutions  of  the  system 


a?  +  ; 


4. 


276  ALGEBRA 

The  solutions  are  ^=l±?i^,   ^^8-1/29       ^^^ 
5  5 

4-21    29     ,._8  +  l/29 
^  5        '    y-       5^' 

The  graphs  of  these  equations  are  the  circle  and  the  straight 
line  PQ  in  Fig.  9. 

The  points  whose  coordinates  are  their  two  solutions  are  the 
points  P  and  Q^  the  points  where  the  graphs  of  the  equations 
intersect. 

Observe  that  the  straight  line  intersects  the  circle  twice.  This 
is  as  tnany  times  as  the  system  has  solutions. 

Example  2.     Interpret  the  solutions  of  the  system 
ix^  +  y'=9, 
x4-l/32/=6. 

The  solutions  are,  x=§,  y=§i   S\  and  x=^,  y=^i/3. 

These  solutions  are  the  same,  i.  e.,  the  system  has  a  pair  of 
equal  solutions 

The  graphs  of  these  equations  are  the  circle  and  the  straight 
line  AB,  Fig.  9.  It  is  seen  that  the  line  AB  does  not  cross  the 
circle  at  all^  but  touches  it  at  one  point  -K,  whose  coordinates 
(I,  |l/3),  are  the  solutions  of  the  system. 

Example  3.     Interpret  the  solutions  of  the  system 

(  icH  2/^=9, 
I  x—2y=S. 

The  solutions  are,   ^^8  +  2i/^19^   ^^-16  +  l/-19.    ^^^^ 


8— 21/-19           -16-1/ -19     -D  4-1       1   4.- 
x= ^   y= -• .    Both  solutions  are  imagmary. 

The  graphs  of  these  equations  are  the  circle  and  the  straight 
line  MN,  Fig.  9.  It  is  seen  that  the  circle  and  the  line  MN  do 
not  intersect  or  touch  at  any  point.  In  general,  ivhen  the  solu- 
tions of  a  system  are  imaginary  expressions.,  the  graphs  of  the 
equations  neither  cross  nor  touch. 


QUADRATIC  AND  HIGHER  EQUATIONS 


277 


Example  4.     Interpret  the  solutions  of  the  system 


This  system  has  four  solutions  : 

a?=T?^l/l3,  2/=Al/l3;  x=-^\\/n,  I/-TJ 

y=  _-^6j|/i3 ;  and  ic=  —  t«^i/13 

The  graphs  of  these 
equations  are  the  two 
ellipses  in  Fig.  10.  The 
four  solutions  are  the  co- 
ordinates of  the  four 
points,  A,  B,  C,  X),  where 
the  curves  meet. 

Example  5,      Interpret  ^ 
the  solutions  of  the  sys- 
J  4xH9?/^=36, 


jl/lo;  X — y^ 


•Vt/13, 


tern 


X^  +  ' 


This  system  has  four 
solutions  : 


x=i\/-15,    y=l\/lO 
x=iV--15,    y=-ii/lO 
x=-ii/-15,    y=iVlO      _ 
and  a?=— f]/  — 15,  y=—^i/l{).     These  solutions  are  imaginary. 

The  graphs  of  these  equations  are  the  ellipse  (1)  and  the  circle 
in  Fig.  10.  It  is  seen  that  the  circle  lies  entirely  within  the  ellipse, 
and  does  not  meet  it  at  all.  This  is  to  be  expected,  since  the 
abscissas  of  the  points  where  they  should  meet  are  imaginary. 

(1) 
c=0.  (2) 

Eliminating  y  from  this  system,  observe  that  we  have  the 
general  quadratic  equation  ax'^  +  bx  +  c  =  0.  Since  all  solutions 
of  the   system  are  solutions  of  the  quadratic  ax'^  +  bx  +  c=0, 


The  graphic  solution  of  the  system  j  ^^  f^'^., 


278 


ALGEBRA 


it  follows  that  the  abscissas  of  the  intersections  of  the  graphs 
of  equations  (1)  and  (2)  will  be  the  solutions  of  the  quadratic. 

Now  since  the  equation  y=x^ 
will  be  the  same  for  all  systems 
formed  from  the  quadratic,  to 
solve  any  quadratic  we  need 
merely  to  find  where  the 
straight  line  ay  +  bx  +  c  =  0  cuts 
the  fixed  parabola. 

Example  6.  Consider  the 
equation  x^—x—2=0.  This  is 
equivalent  to  the  system 

I  y-x-2=0. 

The  graphs  are  (a)  and  (&),  Fig. 

11.     The  abscissas  of  their  points 

of   intersections  are  —1,  and   2, 

the  solutions  of  the  quadratic. 

Example  7.  To  solve  the  qua- 
dratic, ic^— 6a?+9=0,  we  find  the 
intersection  of  the  line  y—Gx  + 
9=0  with  this  same  parabola. 
The  straight  line  touches  the 
parabola  at  the  point  x=3  ; 
hence,  the  solutions  are  3  and  3. 
It  is  thus  seen  that  to  solve  any  quadratic  we  have  simply  to 

find  the  graph  of  ay  +  bx  +  c=0,  the  parabola  remaining  fixed 

for  all  quadratics. 

In  using  this  method  make  on  coordinate  paper,  ruled  to 
centimeters  and  millimeters,  a  perfect  graph  of  y=x'^  for  your 
fixed  parabola.  Now  find  two  points  on  the  graph  otay  +  bx-\-c=0. 
Connecting  these  by  your  ruler  find  the  abscissas  of  the  points 
where  the  ruler  crosses  tlie  parabola,  and  these  abscissas  will  be 
the  solutions  of  the  given  quadratic. 


-1— 

Y 

:  1 

.  1 

qi 

'              L 

J 

.            I 

r-            i            - 

1 

I 

:    T 

f            . 

it    _ 

i  ^2 

t 

4      2_ 

Y 

I._Z    . 

T 

t^      - 

^ 

_^K 

I 

2 

t 

71 

V 

t 

'V           Zt 

2 V 

2 

0 

(tjZ 

z 

)• 

Fig.  11. 


QUADRATIC  AND  HIGHER  EQUATIONS 
The  solution  of  the  system  }  p  ^'^2  +  ^^  _|_  ^^ 


279 


Here  again  the  elimi- 
nation of  y  gives  the 
quadrati  c,  ax^  +  ^aj  +  c  = 
0.  Hence,  the  abscissas 
of  the  intersections  of 
the  graph  of  2/=0, 
which  is  the  x — axis^ 
with  the  parabola^ 
y  —  ax^ -^hx-\- c,  will  be 
the  solutions  of  the  qua- 
dratic equation. 

Example  8.  Solve  the 
equation  a;'-^  ^  4x'  +  3 = 0 . 

The  graph    of   y=x^— 

4a? +  3  is  the  parabola  of 

Fig.    12  which  is   cut  by^ 

the  x—axis,   or  y=0^  in 

points    x=l    and     x=3 

which    are    solutions    of 
Y 
*.     .g  the  quadratic. 

Note. — The  method  of  the  preceding  system  is  preferable  to  this 
one,  since  by  the  method  of  this  system  a  new  parabola  must  be 
found  for  each  equation. 

It  is  evi(ient  that  the  graphic  solution  of  equations  may  be 
employed  with  equations  of  degree  higher  than  the  second. 

Example  9 .     To  sol ve  x'' — 3x  +  2  ^  0 . 

To  solve  this,  we  may  solve  the  system 

(  2/=^-3^  +  2,  (1) 

t  2/=0.  (2) 


280  ALGEBRA 

Some  solutions  of  (1)  are  as  follows  I:*  > 

y 


^ 


ik 


x=  — 3, 
i=-16; 


x=l, 
y=0\ 


x=—2, 
y=0; 

x=0, 

y=2\ 


j  x=S, 

I  y=2o 


X 


20. 

From  these  we  get  the 
graph  in  Fig.  13.  The 
x—axis  cuts  this  graph  at 
the  point— 2  and  touches  it 
at  the  point  1.  Hence,  the 
three  solutions  of  the  equa- 
tion are  —2,  1,  and  1. 


Y  Note. — In    this    and     pre- 

Fig.  13.  ceding    chapters  it  has  been 

seen  that  an  equation  in  two 
unknowns  may  represent  some  kind  of  line,  straight  or  curved,  and 
that  the  solution  of  a  system  of  such  equations  may  be  obtained  by 
carefully  plotting  the  curves  and  measuring  the  distances  of  theii- 
points  of  intersection  from  the  two  axes.  Since  the  accuracy  of  tlie 
solutions  depends  upon  the  accuracy  with  which  the  curves  are  con- 
structed and  the  accuracy  with  which  the  coordinates  of  the  points  of 
intersection  are  measured,  the  graphic  method  is  useful  chiefly  in  dis- 
cussing the  nature  of  the  solutions  rather  than  in  determining  the 
actual  solutions. 

That  an  algebraic  equation  may  represent  a  curve  was  first  discov- 
ered by  Descartes  in  1637.  The  subject  of  Analytical  Geometry  is 
founded  upon  his  discovery  and  is  a  discussion  of  curves  by  use  of  the 
equations  which  they  represent. 


QUADRATIC  AND  HIGHER  EQUATIONS  281 

EXERCISE  87. 

Draw  the  graphs  of  the  equations : 

1.  x'  +  i/'  =  16.  4.  a;'^  =  8y  +  l.  7.  x'-f  =  ^. 

2.  Wx'+f  =  16.  5.  2/^  =  8iK  +  l.  8.  x'-\-^x  +  f  =  S, 

3.  4ic2  +  25/  =  100.       6.  x'  =  ii/  +  4.  9.  x'-^xy-2y'  =  0. 

Draw  the  graphs  of  the  following  systems,  and  interpret 
their  solutions  : 

10.    |"+'^r^'  11.    |»^  +  .'/=25,       ,2^    |-^+.V'=4. 

^'^'    |i«-22/  =  4.  ■^*'*    |a;^  +  36/-36. 

(4a^^  +  V  =  36,  .y     (2/^-2^-1, 


14, 
16, 


f  a;'+a;y-6y2  =  0,  ^g      ( £c^  +  16y^  =  16, 


lQx'  +  i/  =  U. 
By  the  use  of  the  graph  find  the  solutions  of 
Tl^  ic^  +  4«  +  4  =  0.        22.  a;2-2a5-4  =  0.       26.^ x'-2x-l  =  0.  X 
"2©»  £c^-2a;-8  =  0.        23.  x'  =  lQ.  26.  a^='-5i«  +  3  =  0. 

21.  a!'^-2a;  +  4  =  0.       ^V  +  £c-l  =  0.         27.  a;''-2a;=0. 

28.  What  is  the  geometric  interpretation  of  an  imaginary 
solution  of  a  system  ? 

29.  What  is  the  geometric  interpretation  of  equal  solutions 
of  a  system  ? 

30.  How  does  the  graph  show  that  in  general  a  system 
which  is  composed  of  a  quadratic  and  a  linear  equation  has 
two  solutions  ? 

31.  How  does  the  graph  show  that  in  general  a  system 
composed  of  two  quadratic  equations  has  four  solutions  ? 


282  ALGEBRA 

184.  Problems  solved  by  systems  involving  quadratics.  Some 
problems  may  be  solved  by  solving  systems  involving  quad- 
ratic equations. 

Example  1.  Divide  9  into  two  parts  the  sum  of  the  squares  of 
which  shall  be  45. 

Let  the  parts  be  represented  by  x  and  y. 

Then  from  the  conditions  of  the  problem  we  get 

x^y=^,  (1) 

and  a^'  +  2/'=45.  (2) 

Equations   (1)   and    (2)   form  a   system   whose    solutions    are 

These  two  solutions  of  the  system  give  only  one  solution  to  the 
problem. 

The  required  parts  are  3  and  6. 

Example  2.  In  running  a  mile  a  drive  wheel  of  a  locomotive 
makes  264  revolutions  fewer  than  a  wheel  of  the  tender  ;  but 
if  the  wheel  of  the  tender  were  2  feet  greater  in  circumference, 
the  drive  wheel  would  make  only  176  fewer  revolutions  in  a  mile. 

Find  the  circumferences  of  the  wheels. 

Let  x=  number  of  feet  in  the  circumference  of  the  drive  wheel, 
and  2/=  number  of  feet  in  the  circumference  of  the  other  wheel. 

Then  in  running  a  mile  the  drive  wheel  makes  revolu- 
tions, and  the  other  wheel  makes  — - —  revolutions. 

Hence,  from  the  first  condition, 

^-5|«=264.  (1) 

If  the  wheel  of  the  tender  were  2  feet  greater  in  circumference, 
it  would  make  — — ^  revolutions  in  a  mile. 

Hence,  from  the  second  condition, 

2/  +  3       X  ' 


QUADRATIC  AND  HIGHER  EQUATIONS  283 

Equations  (1)  and  (2)  form  a  system  whose  two  solutions  are 
x=20,  y=10  ;  x=-7^,  y=-'^2. 

From  the  nature  of  the  problem  the  solutions  must  be  positive. 
Hence,  the  second  solution  of  the  system  must  be  discarded. 

Therefore,  the  circumference  of  the  drive  wheel  is  20  feet,  and 
the  circumference  of  the  other  wheel  is  10  feet. 

EXERCISE  88. 

1.  Find  two  numbers  such  that  their  sum  shall  be  13,  and 
the  sum  of  their  squares  97. 

2.  Find  two  numbers  such  that  the  sum  of  their  squares 
shall  be  146,  and  the  difference  between  their  squares  96. 

3.  Divide  8  into  two  parts  such  that  the  sum  of  their  cubes 
shall  be  152. 

4.  Find  two  numbers  such  that  their  difference  shall  be  4, 
and  the  difference  of  their  cubes  208. 

5.  Find  two  numbers  such  that  the  square  of  the  greater 
shall  exceed  the  square  of  the  less  by  64,  and  such  that  the 
square  of  the  less  shall  exceed  twice  the  greater  by  16. 

6.  The  sum  of  two  numbers  multiplied  by  their  product  is 
160  ;  and  their  difference  multiplied  by  their  product  is  96. 
Find  the  numbers. 

7.  The  sum  of  two  numbers  is  58;. and  the  sum  of  their 
square  roots  is  10.     Find  the  numbers. 

8.  The  sum  of  two  numbers  equals  the  difference  of  their 
squares;  and  their  product  exceeds  twice  their  sum  by  8. 
Find  the  numbers. 

9.  The  difference  of  the  fourth  powers  of  two  numbers  is 
255  ;  and  the  sum  of  their  squares  17.     Find  the  numbers. 

10.  If  a  number  of  two  digits  be  multiplied  by  its  ones' 
digit,  the  product  will  be  124.     If  the  digits  be  interchanged 


284  ALGEBRA 

and  the  resulting  number  multiplied  by  its  ones'   digit,  the 
product  will  be  156.     Find  the  number. 

11.  Find  the  number  of  two  digits  which  equals  the  square 
of  the  ones'  digit,  and  which  also  equals  4  times  the  sum  of  its 
digits. 

12.  If  the  suni  of  the  squares  of  two  numbers  be  divided  by 
the  first  number,  the  quotient  Avill  be  11  and  the  remainder  6. 
The  first  number  exceeds  the  second  by  6.     Find  the  numbers. 

13.  The  area  of  a  rectangle  is  36  square  inches.  If  its  length 
be  increased  by  3  inches  and  its  width  by  2  inches,  its  area  will 
be  doubled.     Find  its  dimensions. 

14.  A  rectangle  is  20  inches  long  and  16  inches  wide.  How 
much  must  be  added  to  its  width  and  how  much  must  be 
taken  from  its  length,  in  order  that  its  area  may  be  increased 
by  22  square  inches  and  its  perimeter  by  2  inches? 

15.  The  hypotenuse  of  a  right  triangle  is  10.  If  one  leg 
be  increased  by  3  and  the  other  leg  by  4,  the  hypotenuse  will 
become  15.     Find  the  sides  of  the  triangle. 

16.  A  rectangular  lot  which  contains  880  square  yards  is 
2.2  times  as  long  as  it  is  wide.  How  mucl\  will  it  cost  to  build 
a  fence  around  it  at  25  cents  a  linear  foot  ? 

17.  A  guy-rope  to  a  derrick  is  attached  to  a  stake  30  feet 
from  the  foot  of  the  derrick.  If  the  rope  were  16|  feet  longer, 
it  would  reach  to  a  stake  53 i  feet  from  the  foot  of  the  derrick. 
Find  the  height  of  the  derrick  and  the  length  of  the  rope. 

18.  A  flower  garden  contains  1000  square  feet,  and  is  sur- 
rounded by  a  path  5  feet  wide.  The  area  of  the  path  is  750 
square  feet.     What  are  the  dimensions  of  the  garden? 

19.  If  the  numerator  of  a  fraction  be  increased  by  1  and  the 
deuominatov  be  diminished  by  1,  the  resulting  fraction  will  be 


QUADRATIC  AND  HIGHER  EQUATIONS  285 

equal  to  the  given  fraction  inverted ;  and  3  times  the  numer- 
ator of  the  given  fraction  exceeds  2  times  the  denominator  by 
3.     Find  the  fraction. 

20.  A  laborer  received  $32.50  wages.  If  he  had  worked  5 
days  longer,  and  had  received  50  cents  a  day  less,  he  would 
have  received  $41.25.  How  long  did  he  work,  and  what  were 
his  wages  a  day  ? 

21.  A  certain  number  of  men  do  a  piece  of  work  in  a  certain 
number  of  days.  If  there  were  2  fewer  men,  it  would  take  4 
days  longer  to  do  the  work ;  and  if  there  were  twice  as  many 
men,  it  would  take  4  days  less  to  do  the  work.  Find  the  num- 
ber of  men  and  the  number  of  days  required  for  them  to  do  the 
work. 

22.  A  grocer  bought  apples  and  potatoes  for  154.  He  sold 
the  apples  for  $36.80  and  the  potatoes  for  $18.70.  He  gained  as 
many  per  cent  on  the  apples  as  he  lost  on  the  potatoes.  How 
much  did  he  pay  for  each  ? 

23.  A  certain  sum  of  money  placed  at  simple  interest  for 
one  year  amounted  to  $265.  If  the  principal  had  been  $50 
more  and  the  i*ate  1%  less,  the  interest  would  have  been  the 
same.     Find  the  principal  and  the  rate. 

24.  Two  men  can  do  a  piece  of  work  in  4|  days.  It  would 
take  one  four  days  longer  to  do  the  work  than  it  would  take 
the  other.  How  long  would  it  take  each  of  them  to  do  the 
work  ? 

EXERCISES  FOR  REVIEW  (V). 


1.  What  is  the  difference  between  a  surd  and  a  rational  ex- 

ession  f 

%,  What  determines  the  order  of  a  surd  ? 


pression  f 


286  ALGEBRA 

3.  When  is  a  surd  in  the  simplest  form?     By  what  prin- 
ciple may  a  surd  be  reduced  to  the  simplest  form  ? 


4.  Simplify   y'x'ip  ;    2 1   1 65* {a —lif\    v^x^ — ^x^y  +  ^xy"^ ; 


y_l_         /a  +  6 


(x—yy  y    a—Q 
5.  What  kind  of  surds  may  be  added  or  subtracted  ? 


6.  Simplify  i/Sa-2VSla'  +  Syl92ab\ 


7.  Add  i/a'x\y^z)\  i/9aV(y  +  2;),  and  3i/16(y  +  s)^ 


8.  Add  tVv  "72,  -ii   i,  and  6i/21i. 

9.  By  what  principle  do  you  change  the  order  of  a  surd  ? 

10.  Change  i/5,  y  11,  i/13,  to  surds  of  the  same  order. 

2d 


3/27^ 


11.  Write  as  an  e7itire  surd  o 

Sx 

12.  Which  is  the  greater,  i  5  or  v  TO"? 

13.  Write  as  one  surd|/^  y  257 

14.  By  what  principle  are  surds  multiplied  ?    Illustrate. 

15.  Find  the  product  of  i/Sx,  y  27x\  and  y9x\ 

16.  Find  the  product  of  Si/i  +  S]/  5  and  Qy2  +  7yW. 

17.  What  is  the  corijugate  of  2  +  |/5  ? 

18.  Give  a  method  for  dividing  by  a  surd.     Illustrate. 
15  +  3i   3 


19.  Simplify 


15-2|/3 


20.  Find  the  value  of  (2i/a=^6)^;  {bx\    96a;«y;    (v  48icy)l 


21.  Find  the  value  of   y^/^lxY^i    i/vV  +  6a;+9; 


l/^lx'yxXx-yY. 


QUADRATIC  AND  HIGHER  EQUATIONS  287 

22.  Simplify: 

(''>     7I/3-5V2-  ^^    Vf^x- 

31/5  +  51   g  (/)  (I'/^^M?)/ 

^  ''      V/5-V3 


^)    2)/n5-3i/63  +  5v"28. 


W 


^/s-!v^^) 


^(f7)3i/f+i/TV  +  4i/^V  W    l/2Xi/3xi/iXl/i.      ^ 

23.  What  is  an  imaginary  number  ?     Illustrate. 

24.  What  is  the  typical  form  of  an  imaginary  number  ? 

25.  Reduce  ]/— 4ic*  to  the  typical  form. 

26.  What  are  the  values  of  the  successive  powers  of  i/— 1? 
How  is  ^/^-i  usually  represented  ? 

27.  Simplify: 


(a)   V- 

-36a^- 

-l/-49a^  +  l/- 

81«^ 

ip)  v~- 

^•1/- 

-4  i/-l(>. 

(^)    (l/' 

-2  +  1 

/-5)(i/-2-i/ 

-5). 

(^0  (1/ 

-12- 

-l/-15)-i/-3. 

28. 

,  Write  the  conjugate  of  Z-\-y  ■ 

-5. 

29.  What  is  a  complex  number  f    Illustrate. 

30.  Square  i/^^S-j/'^^. 

31.  Divide  3  +  |/^=5  by  4-i/^=^. 

1/^^+3 

32.  Simplify  ^-^_^^-=^' 

33.  How    many   roots  has    a   quadratic    equation    in    one 
unknown  ?    What  kind  of  numbers  may  they  be  ? 

34.  Give  an  example  of  a  2yf(re  quadratic  equation  in  a. 


288  ALGEBRA 

35.  How  do  you  solve  any  pure  quadratic  ?  How  is  the 
solution  of  any  equation  checked? 

36.  Solve  and  check  : 

{a)  4£c^-3  =  0.       {h)   3«2  =  9a2-54.       (c)  ^{t^\)-t{t-\)=^t. 

37.  What  is  a  complete  quadratic  equation?  Give  two 
general  methods  of  solving  such  an  equation. 

38.  Solve  by  factoring  : 

{a)  2a;^  +  9a;-5.  {h)  6y^  +  5y-6  =  0.  (c)   1-\%x  =  ^x\ 

39.  Solve  by  completing  the  square.  (Always  check  your 
solutions.) 

(a)  i«^-4£c  =  32.       {h)  2ic^  +  9;K-5  =  0.        (c)  SGaj'^-aea.  +  S^O. 

40.  Solve  by  the  quadratic  formula  : 

{a)    8a;^  +  2i«  =  3.  (c)    3ic2  +  35-22a;  =  0. 

{h)    x^-\x=\.  (d)  4x'  +  17x=U. 

Solve  the  following  for  the  general  number : 

41    J_  =  _L+^      '      42  ^zL^.?:z8=_?^+l     \\ 

•  a-\     a-2^a-'f  ^-  x+3^  x-3     x^-9^'2' 

■  y-\       2y  ■ 

44.  In  changing  a, /*r«c^/o/2a/  equation  to  an  integral  equation, 
by  what  expression,  in  general,  must  the  members  of  the 
equation  be  multiplied  ? 

45.  How  can  you  tell  the  nature  of  the  roots  of  a  quadratic 
without  solving  the  equation?  What  is  meant  by  the  dis- 
criminant of  a  quadratic  ? 

46.  Solve  ax^-^hx^c  =  ^  and  give  the  relations  that  must 
exist  among  the  coefficients  for  the  different  kinds  of  roots. 

47.  What  must  be  the  value  of  m  if  the  roots  of  ^tx^—lx-^m 
=  0  are  equal  ? 


QUADRATIC  AND  HIGHER  EQUATIONS  289 

48.  What  relations  exist  between  the  roots  and  coefficients 
of  x''+px  +  q  =  0  ? 

49.  Construct  an  equation  whose  roots  are  3  and  —  ^.     One 
whose  roots  are  l  +  j/f  and  1  — i  |. 

50.  Explain  the  method  of  stating  and  solvmg  a  problem  by 
means  of  an  equation. 

51.  Can  you  solve  an  equation  of  a?i(/  degree  ?     What  special 
kinds  of  equations  of  higher  degree  can  you  now  solve  ? 

52.  Solve  the  following  : 
(a)  x*^Q  =  bx\ 


53.  How  many  solutions  has  an  equation  of  the  4th  degree 
in  one  unknown  ?  One  of  the  5th  degree  ?  One  of  the  nth 
degree  ? 

54.  What  is  an  irrational  equation  in  a;  ? 

55.  Give  the  general  method  of  solving  an  irrational  equation. 

56.  When  is  a  solution  said  to  be  introduced  in  a  derived 
equation  ? 

57.  Show  that,  in  general,  when  both  members  of  an  equation 
are  squared,  7ieia  solutions  are  introduced? 

58.  Solve  and  check  : 

(a)  '6\/x=i/xTS  +  VxTQ. 


{b)  1/7 -a;  4-1  =  1/205  + 


l-l/l+a;     i/l-a; 

^''^  i+i/rT^~i/rF^* 

59.  How  many  cube  roots  has  a  number?    How  nmny  fourth 


290  ALGEBRA 

roots?     How  many  nth  roots?     Find   the   cube   roots   of   1. 
Of  8.     Of  27. 

60.  How  many  solutions,  in  general,  has  a  system  of  one 
linear  and  one  quadratic  equation  ?  A  system  of  two  quadratic 
equations  ?     How  is  this  illustrated  by  the  graphs  ? 

61.  What  is  a  defective  system?     Illustrate. 

62.  What  represent  the  solutions  in  the  graphs  of  a  system 
of  two  quadratic  equations  ? 

63.  Solve  the  system  {  f +/;t^lr'' 

Cr 

64.  The  area  of  a  circle  is  equal  to  -^,  where    G  represents 

the  number  of  units  in  the  circumference,  and  r  the  number  of 

Cr 
units  in  the  radius.     From  ^=_-,  and  (7=2;:r  find  ^  in  terms 

of  r  and  r. 

Sr 

65.  From  F=-q-j  ^==4-^^,  and  r=\I>^  find  V  in  terms  of  r 

o 

and  TT ;  also  in  terms  of  U  and  r. 


CHAPTER   XVIII. 
INEaUALITIES. 

185.  Definitions.  If  a— ^  is  positive,  a  is  said  to  be  greater 
than  h.     If  a— ^  is  negative,  a  is  said  to  be  less  than  b. 

The  symbol  for  ''Hs  greater  than^^  is  >,  and  the  symbol  for 
"is  less  than^^  is  <. 

Thus,  a>6  is  read,  "a  is  greater  than  6."  And  a<&isread, 
"  a  is  less  than  6." 

From  the  above  definitions  it  follows  that  when  a  —  h  is 
positioe^  ay>b,  and  when  a—b  is  negative,  a<^b. 

Thus,  since  6—4=2,  therefore  6>4;  since  5— 8=— 3,  therefore 
5<8. 

The  statement  that  two  expressions  are  not  equal,  i.  6.,  that 
one  of  the  expressions  is  greater  or  less  than  the  otlier,  is  called 
an  inequality.  The  expression  at  the  left  of  the  inequality 
sign  is  called  the  first  member  of  the  inequality,  and  the  expres- 
sion at  the  right  of  the  inequality  gign  is  called  the  second 
member. 

Two  inequalities  which  have  the  same  sign  of  inequality  are 
called  inequalities  of  the  same  species.  Two  inequalities  which 
have  opposite  signs  of  inequality  are  called  inequalities  of 
opposite  species. 

Thus,  15>12  and  7>— 2  are  of  the  same  species ;  and  6>— 4 
and  9<17  are  of  opposite  species. 

In  this  chapter,  the  letters  used  in  the  members  of  an 
inequality  represent  07ily  positive  real  nimibers.  A  negative 
number  will  be  denoted  by  a  negative  sign. 

291 


292  ALGEBRA 

186.  Principles  of  inequalities.  The  following  are  some 
useful  principles  of  inequalities. 

(1)  If  equal  numbers  be  added  to,  or    subtracted  from,   the 
members  of  an  inequality,  the  result  will  be  an  inequality  of  the 
same  species  ;  that  is,   . 
if  fl>6,  then  a-\-c^b^c,  and  a—d^b—d. 

For,  since  a> b,  then  a  —  b\&  positive.  Hence,  (a  +  c)  —  (6  +  c), 
which  equals  a  —  b,  is  also  positive.     Therefore, 

a  +  c>^»  +  c.  §  185. 

In  like  manner,  a—d^b—d. 

Evidently  the  proof  would  have  been  similar,  had  the  given 
inequality  been  a<b. 

Thus,  since  10>7,  therefore  10  +  6>7  +  6,  and  10-4>7-4. 

{2)  If  the  members  of  an  inequcdity  be  tnidtiplied,  or  divided, 
by  equal  positive  numbers,  the  result  ivill  be  an  inequality  of  the 
same  species  ;  that  is, 

if  a>6,  then  ac^bc,  and->-. 

c     c 

For,  since  a>^,  therefore  a  — ^  is  positive.     Hence,  c(a  —  b), 

or  ac  —  bc,  \s  positive  when  c  is  positive.     Therefore, 

acybc.  §  185. 

In  like  manner,   ->-,    when  c  is  positive. 

Thus,  if  o  +  2>4,then  multiplying  both  members  by  3,  we  have 

i»  +  6>  12.     And  if  ocf  -f  2xy  Qx,  then  dividing  both  members  by  a?, 
we  have  aj  +  2>6. 

(3)  If  the  corresponding  members  of  tivo  inequalities  of  the 
same  species  be  added,  the  residt  will  be  an  inequality   of  the   same 
species  ;  that  is, 
if  fl>  b,  and  c> (/,  then  a  +  c>  6  -h  (/. 


INEQUALITIES  293 

For,  since  a>^,  and  c>r/,  therefore  a  —  h  and  c—d  are  both 
positive.  Hence,  their  sum,  a—  b  +  c—d,  is  positive ;  i.  e., 
(a-\-c)  —  (b  +  d)  is  positive.     Therefore, 

a  +  c>i  +  (^.  §185. 

Thus,  if  2a?— 2/>10,  and  x+y^4^  by  adding  them  we  have 

3x>U. 

(4)  If  the  members  of  an  inequality  be  subtracted  from  the 
members  of  an  equation^  the  result  loill  be  an  inequality  of  oppo- 
site species  ;  that  is, 

if  a>6/  and  c  =  d,  then  c—a<id—b. 

For,  since  a^b,  and  c  =  d,  therefore  a  —  b  is  2)ositive  and  c  —  d 
is  zero.  Hence,  (c—d)  — (a—b)  is  negative  /  i.e:^  (c—a)  —  (d—b) 
is  negative.     Tlierefore, 

c—a<^d-b.  §  185. 

Thus,  since  15>9,  therefore  20-15<20-9.  • 

(5)  If  the  members  of  an  inequality  be  subtracted  from^  or 
divided  by  the  members  of  another  inequality  of  the  same 
species.,  the  residt  is  not  necessarily  an  inequality  of  the  same 
species ;  that  is, 

if  fl>6  and  c>(/, 

then  a—c  may  or  may  not  >6— (/ 

and  -  may  or  may  not    >;^. 

C  Q 

Thus,  8>4  and  7>2,  but  8-7<4-2,  and  |<|. 
Also,  8>4  and  Q>2,  but  8-6=4-2  and  |<|. 

{6)  If  the  members  of  an  inequality  be  multiplied^  or  divided., 
by  equal  negative  numbers.,  the  residt  will  be  an  inequality  of 
opposite  species  ;  that  is, 

if  fl>6,  then  ac<bc,  and  -<-/  when  c  is  negative. 

c     c 


294  ALGEBRA 

For,  since  a>J,  therefore  a—b  is  jyositive.     Hence,  ac—bc  is 
negative  when  c  is  negative.     Therefore, 

ac<,bc.  §  185. 

Similarly,  -<-• 

Thus,  since  9>6,  therefore  9(-2)<6(-2),  i.e.,  -18<-12. 

(7)  If  the  signs  of  all  terms  of  an  inequality  be  reversed.,  then 
the  symbol  of  inequality  must  also  be  reversed 
This  follows  directly  from  (6)  when  c  equals  —1. 
Thus,  if  3ic-22/  +  6>a-46,  then  2y-^x-Q<4.b-a. 

Note. — These  are  but  a  few  principles  of  inequalities.     There  are 
many  others  which  could  be  established  by  similar  processes. 

187.  Identical  and  conditional  inequalities. 

Inequalities  which  are  true  for  all  values  of  the   general 
numbers  involved  are  called  identical  inequalities. 

Thus,  a  +  10>a  is  an  identical  inequality. 

Inequalities  which  are  }iot  true  for  all  values  of  the  general 
numbers  involved  are  called  conditional  inequalities. 

Thus,    6a?  +  2>0   is   true   only  under   the  condition  that  x  is 
greater  than  —^. 

Note. — It  is  to  be  observed  that  identical  and  conditional  inequalities 
are  analogous  respectively  to  identical  and  conditional  equations. 

188.  Proofs  of  some  identical  inequalities.      Some  identical 
inequalities  may  be  established  by  use  of  the  pJ-inciples  in  §  186. 

Example  1.     Show  that  a'^  +  b^^2ab,  if  a  and  b  are  unequal. 
Since  a  and  b  are  unequal, 
therefore,  {a—byy>0, 

or  a'-2ab-i-b'>0. 

Adding  2a6,  a'  +  b'>2ab.  §186,  (1) 


INEQUALITIES  296 

Example  2.     If  a  and  h  are  unequal,  and  a  +  6>0,  show  that 

We  have  a^  +  h''>2db.  Ex.1. 

Subtracting  a6,  a^-ab  +  W^ah.  §186,  (1) 

Multiplying  by  a +  6,  d'-\-lf^a'h^ah\  §186,  (2) 

Example  3.  The  sum  of  any  positive  number  n  and  the  quo- 
tient -  is  greater  than  2. 

We  have  a'  +  h'''>2db.  Ex.1. 

Now  let  d^  be  n  and  let  If  be  -. 

n 

Then,  substituting  in  the  above  inequality,  we  have 

-  »  +  i>2- 

189.  Solving  conditional  inequalities.  A  conditional  in- 
equality is  said  to  be  solved  when  the  possible  values  of  the 
unknown  numbers  which  will  satisfy  it  are  found.  The  range 
of  these  values  is  discovered  by  means  of  principles  such  as 
those  established  in  §  186. 

Example  1,     Find  the  possible  values  of  x  in 

37-2^       ^  3x-8     ^ 
3— +  a'> ^-9. 

Multiplying  by  12,  148-8^+ 12a;>9x-21-108,  §  186,  (2). 
or  148 +  4;r>  9^-132. 

Subtracting  148  + 9aj,  -5ic>— 280.  §186,(1). 

Dividing  by  -5,  x<m.  §  186,  (6). 

Hence,  the  inequality  is  true  for  all  values  of  x  less  than  56. 

Example  2.    Find  the  possible  values  of  x  in  x^'>Qx+16. 

Subtracting  6ic  + 16,  x''-6x-16>0.  §186,(1). 

Factoring,  (x-8){x  +  2)>0. 

Hence,  the  factors  a?— 8  and  x  +  2  must  both  be  positive  or  both 
be  negative, 


296  ALGEBRA 

To  make  both  positive,  x  must  be  greater  than  8. 
To  make  both  negative,  x  must  be  less  than  —2. 
Hence,  a^>8  or  <— 2. 

Example  3.     Find  what  values  of  x  will  satisfy  the  system 
10a?>3x  +  49,  (1) 

a7+5<|+55.  (2) 

From  (1),  x>7. 

From  (2),  x<75. 

Hence  the  system  is  satisfied  by  any  value  of  x  between  7  and  75. 

Example  4.     Find  what  values  of  x  and   y  will  satisfy  the 
system 

\  3j!  +  i/>10, 

\x^y=Q, 

Subtracting  (2)  from  (1),  2x>2. 

Whence,  ic>l. 

Multiplying  (2)  by  3,  3a: +3?/= 24. 
Subtracting  (3)  from  (1),  -2i/>  -14. 

Dividing  by  —2,  2/<7. 

Hence,  any  solution  of  (2)  in  which  x>l  and  y<i7  will  satisfy 
both  (1)  and  (2). 

EXERCISE  89. 

Establish  the  following  identical  inequalities  in  which  the 
letters  represent  unequal  positive  numbers  : 

ik-\-y     2xy  ^.  m^n  +  mn^^^m'^n^. 

3.  a  +  5>2i/aJ.  6.  a*  +  b*:^a'b\ 

Suggestion  :    Let  a=x^  and  h=y'\ 


(1) 

(2) 

§186, 

(1). 

§186, 

(2). 

(3) 

§186, 

(1). 

§186, 

(6). 

INEQUALITIES  297 

Find  what  values  of  x  will  satisfy  the  following: 

7.  2aj-3>7.  11.  a;2-2a;>3. 

8.  ^-hx<:^x-l\.  12.  _l_<a;  +  2. 

9-  :-±|>3-  13.      6    _    3  8 

a?— 0     ic— 4     tc  — 3 

10.  -1-  <_L  14.  r^^a<^_\ 

Find    the   values    of    x    that   will    satisfy    the    following 
systems  : 

..      (3ii;-5<2a;+l,  r3a;-16     5 

^^-    l3a^  +  4>x  +  8.  ^g     I— ^<3' 

17     (3(0^-2X2(0^-3),  r4^     ^      a;       ^ 

l2a3-7>5.^+5.  !^l>^=r2  +  3, 

.„     j^(^  +  3)>ic(i«-5)  +  16,  1  2a;       ,  p^   4x 

^°-    |3a;(a;  +  2)<3a;(a;-l).  L.'«  +  3'^'^^a;  +  7' 

Find  the  values  of  a?  and  y  that  will  satisfy  the  following 
systems : 

21     ji«+2/=10,  ric  +  2 


24 

22.    ^  3a;  +  2/>14, 


3"  +4y>2, 

y  +  11     ^+1^-, 

11  2 


|a;  +  2y=13. 

(7a;  +  4y  =  l,  ,  __  _ 


23.    Il^  +  f^^i'  25.    |?^  +  *y>l. 


CHAPTER  XIX, 

RATIO  AND  PROPORTION. 

190.  The  Ratio  of  one  number  to  another  number  of  the 
same  kind  is  the  quotient  obtained  by  dividing  the  first  num- 
ber by  the  second.  It  follows  from  the  definition  that  the 
ratio  of  two  numbers  is  an  abstract  numher  indicating  the  num- 
ber of  times  one  contains  the  other^  or  the  part  one  is  of  the 
other. 

Thus,  the  ratio  of  6  to  2  is  6-^2,  or  8.     The  ratio  of  $8  to  $4  is  2. 
The  ratio  of  5  ft.  to  8  ft.  is  5-i-8,  or  |.     The  ratio  of  a  to  Z>  is  a-=-6, 
•    a 

OV   y 

It  follows  that  an  indicated  ratio  is  a  fraction.  Hence,  a 
ratio  may  be  expressed  by  any  of  the  signs  used  to  express  a 
quotient  or  a  fraction. 

Thus,  the  ratio  of  3  to  4  may  be  written  3^4,  3/4,  |,  or  3:4. 

In  the  ratio  between  two  numbers  the  dividend,  or  numera- 
tor, is  called  the  antecedent,  and  the  divisor,  or  denominator,  is 
called  the  consequent.  The  two  together  are  called  the  terms 
of  the  ratio. 

The  ratio  of  a  to  b  is  sometimes  called  the  direct  ratio  of  a 
to  b.  And  the  ratio  of  J  to  «  is  called  the  inverse  ratio  of  a 
to  b. 

Thus,  the  direct  ratio  of  3  to  7  is  f .  The  inverse  ratio  of  3  to  7 
is  |.  The  inverse  ratio  of  one  number  to  another  is  the  direct 
ratio  inverted. 

298 


RATIO  AND  PROPORTION  299 

191.  It  is  clear  from  the  definition  of  a  ratio  that  all  of 
the  laws  established  for  fractions  must  apply  to  ratios. 

Ratios  may  be  reduced  to  higher  or  lower  terms.  They  may  be 
added,  subtracted,  multiplied,  divided,  raised  to  powers,  and  have 
roots  extracted.  Ratios  may  be  compared  by  reducing  them  to  a 
common  denominator,  and  comparing  the  resulting  numerators. 

Ratios  are  compounded  by  taking  their  product. 
Thus,  the  ratio  compounded  of  |  and  f  is  -^%. 

192.  Commensurable  and  incommensurable  numbers. 

Two  numbers  whose  ratio  can  be  exactly  expressed  by  two 
whole  numbers  are  called  commensurable  numbers,  i.  e.,  two 
numbers  are  commensurable  when  there  can  be  found  some 
third  number  of  the  same  kind,  called  their  common  measure^ 
that  is  contained  an  integral  number  of  times  in  each.  Two 
numbers  whose  ratio  can  not  be  exactly  expressed  by  two  whole 
numbers,  i.  6.,  two  numbers  that  have  no  common  measure^  are 
called  incommensurable  numbers. 

Two  fractions  are  commensurable  if  their  numerators  and 
denominators  are  commensurable.  For  the  ratio  of  two  such 
fractions  can  be  expressed  as  the  ratio  of  two  whole  numbers. 

Thus,  the  ratio  of  |  to  A=l-A-|-V-=!|. 

If  the  ratio  of  two  numbers  is  a  surd,  the  numbers  are  incom- 
mensurable. 

Thus,  the  ratio  of  v  2  to  V^=y-^  =  ^—^-    The  ratio  can  not 

be  expressed  by  two  whole  numbers,  hence  \   2  and  i/5  are  m- 
commensurdble. 

The  ratio  between  two  incommensurable  numbers  is  called 
an  incommensurable  ratio. 

193.  Proportion.  A  proportion  is  an  equation  each  of 
whose  members  consists  of  a  ratio.     Four  numbers,  a^  b,  c,  d. 


300  ALGEBRA 

are  said  to  be  in  proportion  when  the  ratio  oi  ato  b  equals  the 
ratio  of  c  to  d. 

The  most  used  forms  in  which  the  proportion  may  be  writ- 
ten are 

(1       c 

T  =  -vand  a  :  b=c  :  d,  sometimes  written  a  :  b::c  :  d. 

This  proportion  is  read  "  a  divided  by  b  equals  c  divided  by 
f?,"  or  "  a  is  to  ^  as  c  is  to  t/." 

The  terms  of  the  ratios  in  a  proportion  are  called  the  terms  of 
the  proportion.  The  first  and  fourth  terms  are  called  the 
extremes,  and  the  second  and  third  terms  are  called  the  means 
of  the  proportion. 

Thus,  in  2:x=Qx'^:Sixf,  2  and  So(^  are  the  extremes,  and  x  and 

6x^  are  the  means. 

a     c 
In  the  proportion  T=-p   d  is  called  the  fourth  proportional  to 

ay  by  and  c. 

194.  In  the  following  sections  are  given  the  most  im- 
portant principles  in  proportion.  The  student  should  master 
these  principles.  •  •      . 

195.  In  any  true  proportion  the  product  of  the  extremes 
equals  the  product  of  the  means. 

Suppose  a  :  b=c  :  d. 

a     c 
Written  in  fractional  form,  ^=^. 

Multiplying  by  bdy  ad^bc, 

which  proves  the  principle. 

Thus,  in  2:  4=6: 12,  2x12=4x6. 

196.  If  the  product  of  two  iiumbers  equals  the  product  of 
two  other  numberSy  then  these  four  numbers  form  a  proportion 
in  ichich  the  extremes  are  the  factors  of  either  product  and  the 
means  are  the  factors  of  the  other  product. 


ad=  be. 

a     c 
b-d' 

c     a 

d     b 

i>  d 

RATIO  AND  PROPORTION  301 

Suppose  that 
Dividing  by  bd, 

or  dividing  by  ac,  -=7:^     or  7, = -j    etc. ; 

which  proves  the  principle. 

Thus,  since  2x10=4x5,  therefore  2:  5=4: 10,  or  5: 10=2;  4. 

It  follows  from  this  principle  that  to  test  the  correctness  of  a 
proportion  it  is  sufficient  to  shoio  that  the  product  of  its  extremes 
equals  the  product  of  its  means. 

Thus,  2  :  x=6a?^  :  Zj?  is  a  correct  or  true  proportion,  for 
2  •  Zd^=x  ■  Qx\ 

197.  Remark.  It  should  be  noted  that  by  the  principle 
of  §  196  we  get  from  the  equation  ad=bc  the  eight  proportions : 

1.  a  :  b  =  c  :  d;  3.  d  :  b  =  c  :  a;  5.  b  :  a  =  d  :  c ; 

2.  a  :  c=b  \  d\  ^.  d  :  c  =  b  :  a;  Q.  b  :  d=a  :  c; 

7.  c  :  a  =  d  :  b;  8.  c  :  d=a  :  b. 

It  should  also  be  noted  that  sii>ceany  one  gi\esad=bc,  from 
any  one  of  them  all  the  others  follow. 

Observe  that  (2)  was  obtained  from  (1)  by  interchanging  the 
means,  and  that  (3)  was  obtained  from  (1)  by  interchanging 
the  extremes.  The  old  mathematical  terms  for  such  changes 
are,  respectively,  mean  alternation  and  extreme  alternation. 

Observe  also  that  (5)  was  obtained  from  (1)  by  interchanging 
the  terms  of  each  ratio.     This  process  is  called  inversion. 

In  (4)  we  observe  that  it  is  obtained  from  (1)  by  inter- 
changing the  means  and  the  extremes. 

Illustrate  each  of  the  8  proportions  given  above  with  numbers. 

198.  The  products  of  the  corresponding  terms  of  two  or 
r>iore  proportions  form  a  proportion. 


302 

ALGEBRA 

Suppose  that 

a     c 

b-7( 

mp 

and 

x_io 
y~lj' 

Multiplying, 

am  X     cpw 
buy     d  q  V 

or 

amx     cpw 
bny      cJqv 

Hence,  amx^  buy, 

cpw,  and  dqi^  are  in  proportion. 

Axiom  3. 


199.  Like  powers  or  like  roots  of  the  terms  of  a  proportion 
are  in  proportion. 


Suppose 

that 

a  :  b  =  c  :  d. 

or 

a     c 
1>-71' 

Then, 

it)- -($)-. 

Axiom  5. 

or 

\  /        \  / 
b""     d"' 

Again, 

n-vi 

Axiom  6. 

n  —         n   — 

\/a     1    c 

or  „  _ 

yb    yd 

Thus,  since  2  :  3=4  :  6,  therefore  2=^  :  ^^=^''  :  6^ 
or  8  :  27=64  :  216. 

And  since  4  :  25=16  :  100,  therefore  |/4  :  |/25=i/T6  :  i  100, 
©r  2  :  5=4  :  10. 

200.  The  terms  of  a  proportioii  are  in  proportion  by  addi- 
tion ;  i.  e.,  the  sum  of  the  first  two  terms  is  to  the  first  (or  second) 
as  the  sum  of  the  last  tico  terms  is  to  the  third  (or  fourth). 


RATIO  AND  PROPORTION  303 


§  195. 


Suppose  that 

a     c 
b-d' 

Then, 

ad— he. 

Adding  bd^ 

ad-^hd=bc-\^bd. 

or 

d(a  +  b)=b(c-i-d). 

Hence, 

a-\-h     c  +  d 

b     d  ■ 

In  like  manner, 

a-{-h     c^d 

§  196. 
§  196. 


a  c 

Thus,  since  3  :  5=12  :  20,  therefore  (3  +  5)  :  5= (12 +  20)  :  20, 
or  8  :  5=32  :  20. 

The  old  mathematical  term  for  this  process  is  composition. 

201.  The   tenuis  of  a  2^^02)ortio)i  are  in  pro2^ortion  by  svh- 

tr action  ;  i.  e.,  the  difference  of  the  first  tico  terms  is  to  the  first 

{or  second)  as  the  diff'erence  of  the  last  tvm  terms  is  to  the  third 

{or  fourth). 

a     c 
Suppose  that  t— ;/• 

Then,  ad=bc.  §  195. 

Subtracting  bd^        ad—bd=bc  —  bd, 
or  d{a  —  b)=  b(c — d) . 

Hence,  ^^JlI^.  §  196. 

T     vi  a—b     c—d 

In  like  manner,  = 

a  c 

Thus,   since    10  :  3=30  :  9,    therefore    (10-3)  :  3=(30-9)  :  9, 

or  7  :  3=21  :  9. 

The  old  mathematical  term  for  this  process  is  division. 

202.  The  terms  of  a  proportion  are  in  proportion  by  addl- 
tion  and  subtraction  ;  i.  e.,  the  sum  of  the  first  two  terms  is  to 
their  differenae  as  the  sum  of  the  last  tico  terms  is  to  their  dif- 
ference. 


304 

ALGEBRA 

Suppose 

that 

a     c 

Then, 

h          d   ' 

And 
Dividine' 

a—h     c—d 
b         d    ' 

a  +  b_c-{-d 

a—o     c  —  d' 


§200. 

§201. 

Axiom  4. 


This  and  the  preceding  sections  will  enable  us  to  obtain  an 
equivalent  proportion  from  a  given  proportion. 

(X      c 

Example.     Suppose  T,~d'  ^^^ 

By  §199,  p=^.  (2) 

By  §  201,  ^JL^.  (3) 

By  §  199, ^ -^ (4) 

203.  Continued  proportion.  When,  in  a  series  of  numbers, 
the  first  is  to  the  second  as  the  second  is  to  the  thirds  and  so 
on,  the  numbers  are  said  to  be  in  continued  proportion. 

Thus,  a,  5,  c,  d,  e,  are  in  coyitinued  proportion,  if  T— 7=^— J' 

In  the  continued  proportion  ^=-,  which  is  called  a  mean  pro- 
portion, b  is  called  a  mean  proportional  between  a  and  c,  and  c  is 
called  the  third  proportional  to  a  and  b. 

Thus,  4: 10=10:  25  is  a  mean  proportion.  10  is  the  mean  propor- 
tional between  4  and  25.  And  25  is  the  third  proportional  to  4  and 
10. 

204.  The  tnean  proportional  between  two  mimhers  equals  the 
square  root  of  their  product. 


RATIO  AND  PROPORTION  305 

Let  b  be  the  mean  proportional  between  a  and  c. 

mu  ah 

Then,  -t  =  - 

*  .  he 

And  }f  =  ac.  §195. 

Hence,  h=\/ao. 


a 

m 

_/>_ 

X 

b 

71 

<1 

y 

a 

=  7*. 

Thus,  in  the  proportion  9: 18=18:  36,  we  have  18  =  1/9 x 36. 

205.  In  a  series  of  equal  ratios  the  sum  of  the  antecedents  is 
to  the  sum  of  the  consequents  as  any  antecedent  is  to  its  con- 
sequent. 

Suppose  that 

Let 

Then  —=^r.~  =  r.-  =  r.  etc.  Axiom  7. 

n       '  q       '  y 

Clearing  of  fractions,         a=rb^  m=rn^  p  =  rq^  x  —  ry^  etc. 
Adding,        a  +  m-fp  +  a;+   •  •  •  •   =rb^rn^rq^ry^r   •  •  •  • 

Dividing  by     b^rn-\-q-^'y^r  •  •  •  •  » 

a-Vm-\p^-x■^- am 

r^"—. , =r=7;  =— =etc. 

o-\r  n-^-q^y^ o      n 

2_4      8      16     2  +  4  +  8  +  16  30 

ihus,  3~6~12~24'-3  +  6  +  12  +  24  ^^  45' 

206.  The  representation  of  a  ratio  by  a  single  letter.,  as  in 
§  205,  is  often  a  useful  process.  In  order  to  test  the  accuracy 
of  a  proportion,  we  may  proceed  as  in  §  196,  or  as  in  the 
example  in  §  202.  But  often  it  is  more  convenient  to  rep- 
resent the  equal  ratios  by  a  single  letter,  and  then  show  that 
the  proportion  reduces  to  an  identity, 

20 


306  ALGEBRA 


Example.     If  -r  =  :^,  show  that     /  ..     ,~  3.  - 

Let  r=r.     Then    ^=r. 
Hence,  a=br^  and  c=dr. 


Substituting,  }AV+£^vV^+£ 

VdV  +  d'    y^d'r^  +  d' 
Factoring  and  reducing, 

b\/?Tl     bV'r^  +  l 
dVr^^Ti~d^^r^ 

or  d~d'  ^^  identity. 

Hence,  the  proportion  is  true. 

207.  Since  a  proportion  is  an  equation,  it  may  be  solved 

for  any  number  in  it. 

Example  1.     a :  b=c :  d.     Find  the  value  of  d. 

a     c 
Writing  in  fractional  form,     x=^' 

Multiplying  by  6d,  ad=bc. 

be 
Dividing  by  a,  ^~a' 

Example  2.     Find  the  fourth  proportional  to  3,  2ic^  and  21a?. 
Let  a  represent  the  fourth  proportional. 

Then.  ^3=-- 

Multiplying  by  2ax^,  Za=42o(?. 

Dividing  by  3,  a=14a?^. 

Example  3.     Find  the  third  proportional  to  a'  and  a6'. 
Let  X  represent  the  third  proportional. 

Then,   •  4=2^. 

'  db^      X 

Multiplying  by  db^x^          a^x=a^b*. 

Dividing  by  a^,  x=b^. 


RATIO  AND  PROPORTION  307 


EXERCISE   90. 


1.  What  is  the  inverse  ratio  of  6  to  11? 

2.  Reduce  the  ratio  of  10  to  18  to  lowest  terms. 
Find  the  simplest  expression  for  the  ratio  of : 

3.  bx  to  Ihx^.  4.   -  to  -T.  5.  7 rr,  to 


a         h'  '  \x—yy        {x—yf 

6.  Which  ratio  is  the  greater,  -J^  or  ||  ? 

7.  Write  in  descending  order  of  magnitude  |,  i|,  f ^. 

8.  Write  the  ratio  co^npounded  of  the  ratios  J,  f^^  |. 

9.  Write  the  ratio  compounded  of  the  ratios  -,  'A,  tt-t^. 

(a\l>V       12 
10.  Write  the  ratio  compounded  of  the  ratios  ^^. — --,    —ry 


11.  What  must  be  the  value  of  a,  if  the  ratio  -— -^  equals 
the  ratio  |  ? 

12.  What  must  be  added  to  each  term  of  I  to  make  it  equal 

to  I? 

13.  Two  numbers  are  in  the  ratio  of  3  to  4,  and  their  sum 
is  35.     Mn4  the  numbers. 

14.  Two  numbers  are  in  the  ratio  of  7  to  5,  and  their  differ- 
ence is  6.     Find  the  numbers. 

15.  Two  numbers  are  in  the  ratio  of  2  to  9,  and  the  difference 
of  their  squares  is  to  the  square  of  their  difference  as  11  is  to  7. 
Find  the  numbers. 


308  ALGEBRA 

Find  i\\Q  fourth  proportional  to  : 

16.  3,  4,  and  36.  18.  p>,  v,  pv.  20.  «,  h,  c. 

17.  X,  ^x\  bx\  19.  1,  7,  10.  21.  x,  y,  z. 
Find  the  third  proportional  to : 

22.  1,  2.  24.  p,  pq.  26.  a,  Z>. 

23.  X,  '6x\  25.  10,  1.  27.  ^,  y. 

Find  the  mean  p)roportional  between  : 

28.  2,  8.  30.  x\  ^x\  32.  a,  h. 

29.  15,  135.  31.  6,  9.  33.  x,  y. 

34.  In  the  proportion  T^p'^  17772'    solve  for  h. 

35.  Show  that  |  and  5  are  com.mensnraMe  numbers. 

36.  Show  that  ^  and  2J  are  commensurable  numbers 

37.  Show  that  7/2  and  \^\^  are  commensurable. 

38.  Show  that  f^'I  and  f/3  are  incommensurable. 

39.  Show  that  ]/15  and  7/5  are  incommensurable. 

40.  Show  that  7  :  12  =  21  :  36  is  a  true  proportion. 
\i  a  :  b=^c  :  (7,  show  that : 

41.  a^  :  b''=-ac  :  M 

42.  a&  :  bd'^  =  a^c  :  ^V?. 

43.  (5a  +  Z>)  :  ^>=(5c  +  (^?)  :  ^?. 

44.  a  :  (a  +  c)  =  (a  +  ^>)  :  {a^b^c^d). 

45.  c  :  d=-f  :  -. 

46.  2a  :  5c  =  2J  :  5r?. 

47.  a  :  b=\a~c  :  1/^. 

48.  {a^b)  :  (c  +  f?)  =  7/aMT'  :  v"eTd\ 


49.  7/c«^-^»=^  :  7/c^-</^  =  7/«^  +  ^>'  :  yc'^d\ 


RATIO  AND  PROPORTION  309 

50.  {a-b)  :  {c-d)=^d^^'  :  ^&^^\ 

51.  {a'c  +  ac')  :  {b'd+bd')  =  {a  +  cf  :  {b^-d)\ 

62.  The  sides  of  a  triangular  field  are  as  4  :  5  :  6,  and  the 
distance  around  it  is  1,200  yards.     What  are  the  sides  ? 

53.  What  must  be  added  to  2,  5,  and  12,  in  order  that  the 
fourth  proportional  to  the  sums  may  be  24. 

54.  Express  the  ratio  of  3^  to  1^  by  the  ratio  of  two  whole 
numbers. 

55.  The  difference  between  two  numbers  is  4,  and  their 
product  is  to  the  sum  of  their  squares  as  6  is  to  13.  Find  the 
numbers. 

56.  A  tree  casts  a  shadow  80  feet  long,  when  a  rod  4  feet 
high  casts  a  shadow  5  feet  long.     How  high  is  the  tree  ? 


CHAPTER   XX. 

VARIATION.    ALGEBRAIC  EXPRESSION  OP  LAW. 

208.  Variables  and  constants.  In  many  problems  in  mathe- 
matics, numbers  are  involved  %chose  values  are  changing. 
Such  numbers  are  called  variable  numbers,  or  variables.  For 
distinction,  numbers  whose  values  do  not  change  during  the 
investigation  of  a  certain  problem  are  called  constants. 

For  example,  one's  age  is  a  variable  magnitude,  expressed  by 
a  'number  which  is  always  increasing.  The  weight  of  a  stone 
lying  on  the  ground  is  a  constant  magnitude,  expressed  by  a 
r>umber  of  pounds  which  does  not  change. 

Two  or  more  variables  may  be  so  related  that  a  change  in 
the  value  of  one  will  cause  corresponding  changes  in  the 
values  of  the  others.  This  relation  between  the  variables  is 
expressed  by  means  of  an  equation. 

Thus,  in  case  of  a  running  train,  the  distance,  time,  and  speed 
are  varnables,  and  their  relation  is  expressed  by  the  equation 
vt=d,  where  v,  ^,  and  d  represent  the  speed,  time,  and  distance, 
respectively. 

If  the  variables,  x  and  y,  are  connected* by  the  equation  y=x^^ 
then  when  x  assumes  the  successive  values  1,  2,  3,  4,  5,  6,  7,  8, 
9,  etc.,  y  assumes  the  corresponding  values  1,  4,  9,  16,  25,  36,  49, 
64,  81,  etc. 

If  one  variable  depends  for  its  values  upon  the  values  of 
another,  the  first  is  called  a  dependent  variable,  and  the  second 
is  called  an  independent  variable. 

810 


VARIATION.     ALGEBRAIC  EXPRESSION  OF  LAW       311 

Thus,  in  the  example  above,  if  x  is  an  independent  variable,  y 
is  a  dependent  variable,  for  as  x  changes  in  value  y  also  changes, 
and  the  value  which  y  assumes  depends  upon  the  value  which 
X  assumes. 

209.  Direct  variation.  If  two  variables  are  so  related  that 
during  all  of  their  changes,  their  ratio  remains  constant^  each  is 
said  to  vary  as  the  other,  or  to  vary  directly  as  the  other.  As 
the  first  increases,  the  second  also  increases  ;  and  as  the  first 
decreases,  the  second  also  decreases. 

If  X  varies  directly  as  y,  their  relation  is  expressed  by  the 
equation 

~=k,  or  x=ky, 

where  ^  is  a  constant. 

The  symbol  oc  is  called  the  siyn  of  variation,  and  is  read 
"  varies  as."  Thus  a  cc  b  is  read  "  a  varies  as  5,"  and  is  equiv- 
alent to  the  equation  t=^,  or  a=kb- 

Example  1.     xcc  y^  and  when  y=4t,  x=20.     Express  the  rela- 
tion between  the  variables. 
Let  x=hy. 

This  equation  must  be  satisfied  by  2/=4  and  x=2Q. 
Substituting,  20=4/b. 

Hence,  A?=5. 

Therefore,  x=5y  is  the  required  expression. 

210.  Inverse  variation.  One  variable  is  said  to  vary  in- 
versely as  another  when,  during  all  of  their  changes,  their  j^roduct 
remains  constant.     As  one  increases,  the  other  must  decrease. 

If  X  varies  inversely  as  y,  their  relation  is  expressed  by  the 
equation  xy=k,  where  ^  is  a  constant. 

Example  1.  Given  that  x  varies  inversely  as  y,  and  when 
a?=2,  2/=4.    Find  the  relation  between  them. 


312  ALGEBRA 

Let  xy—k. 

This  equation  must  be  satisfied  by  ic=2,  2/=4. 

Substituting,  2-4=A;. 

Hence,  fc=8. 

Therefore,  xy=8  is  the  required  equation: 

211.  Joint  variation.  One  variable  is  said  to  vary  as  two 
others  jointly  when  the  first  varies  directly  as  the  product  of 
the  other  tico. 

If  X  varies  as  y  and  z  jointly^  their  relation  is  expressed  by 
the  equation  x=kyz. 

Example  1.     Given  that  a  varies  as  h  and  c  jointly,  and  when 
a=36,  6=3  and  c=2.     Express  the  relation  between  the  variables. 
Let  a=kbc. 

This  equation  must  be  satisfied  by  a=36,  &=3,  c=2. 
Substituting,  36=:fc-3-2. 

Hence,  k=6. 

Therefore,  a=66c  is  the  required  equation. 

212.  One  variable  is  said  to  vary  directly  as  a  second  and 
inversely  as  a  third,  when  it  varies  as  the  quotient  of  the  second 
divided  by  the  third. 

If  t  varies  directly  as  d  and  inversely  as  y,  the  relation  be- 
tween them  will  be  expressed  by  the  equation  ^=Z;-,  where 
^  is  a  constant. 

Example  1.  Given  that  t  varies  directly  as  d  and  inversely  as 
r,  and  when  f=20,  d=8,  and  v=2.     Find  d  when  t=\i)  and  v=^. 

Let  t=k  -■ 

V 

This  equation  is  satisfied  by  f =20,  cZ=8,  u=2. 
Substituting,  20=|A?. 

Hence,  k=5. 


VARIATION.    ALGEBRAIC  EXPRESSION  OF  LAW      313 
Therefore,  the  equation  becomes 

When  ^=10  and  i;=3,  this  equation  becomes 

10=  sf. 
Hence,  d=6. 

EXERCISE  91. 

1.  fcocy,  and  when  2/==5,  x=lb.  Write  the  equation  be- 
tween tliem. 

2.  tcoc  2/,  and  when  2/  =  2,  x=VI.     Find  x  when  y  =  l. 

3.  X  varies  inversely  as  y,  and  when  2/  =  3,  a;  =  8.  Write 
the  equation  between  them. 

4.  X  varies  inversely  as  y,  and  when  i/^lO,  £c=5.  Find  a; 
when  2/ ==2. 

5.  X  varies  jointly  as  y  and  ^,  and  when  2/  =  5,  and^=l, 
£c=40.     Write  the  equation  between  them. 

6.  X  varies  jointly  as  y  and  z^  and  when  y  =  ^  and  2  =  5, 
05=150.     Find  z  when  a;  =  120  and  2/  =  6. 

7.  X  varies  directly  as  y  and  inversely  as  ^,  and  when  y=l^ 
and  2=2,  93  =  24.     Find  y  when  a:;  =  14  and  2;  =  2. 

8.  «^cc2/^  and  when  2/^4,  £c  =  16.  Find  the  equation  be- 
tween X  and  2/. 

9.  The  velocity  of  a  body  let  fall  toward  the  ground  varies 
as  the  time  during  which  it  has  fallen  from  rest,  and  the 
velocity  at  the  end  of  3  seconds  is  96  feet  per  second.  Write 
the  equation  between  the  velocity  and  time. 

10.  The  area  of  a  triangle  varies  jointly  as  the  altitude  and 
base.  When  the  altitude  is  4  inches  and  the  base  9  inches, 
the  area  is  18  sq.  inches.  What  will  be  the  area  when  the 
base  is  6  inches  and  the  altitude  10  inches  ? 


3l4  ALGEBRA 

11.  The  area  of  a  circle  varies  as  the  square  of  its  radius. 
When  the  radius  is  6  feet  the  area  is  113.0976  square  feet. 
What  is  the  area  when  the  radius  is  8  feet  ? 

12.  The  vohime  of  a  gas  varies  inversely  as  the  pressure 
upon  it.  When  the  pressure  is  8  lbs.,  the  volume  is  8  cubic 
inches.     What  is  the  volume  when  the  pressure  is  4  lbs.? 

13.  The  distance  a  body  falls  from  rest  varies  as  the  square 
of  the  time.  In  2  seconds  it  falls  64  feet.  How  far  will  it  fall 
in  4  seconds  ?    How  far  will  it  fall  during  the  fourth  second  ? 

14.  Tlie  number  of  oscillations  made  by  the  pendulum  of  a 
clock  in  a  given  time  varies  inversely  as  the  square  root  of  its 
length.  A  pendulum  39.1  inches  long  makes  one  oscillation 
in  a  second.  How  long  must  a  pendulum  be  to  make  4  oscil- 
lations in  one  second  ? 

15.  In  playing  see-saw  the  two  persons  must  sit  on  the 
board  at  distances  from  its  point  of  support  which  vary 
inversely  as  the  weights  of  the  persons.  Where  must  the 
support  be  placed  under  a  board  16  feet  long  in  order  that  two 
girls  Aveighing  respectively  65  lbs.  and  80  lbs.  may  play  see-saw? 

THE  ALGEBRAIC  EXPRESSION  OP  LAW. 

213.  Law  expressed  by  the  equation.  An  equation  contain- 
ing two  or  more  variables  expresses  a  law  according  to  which 
the  variables  change  their  values.  Since  there  is  a  relation 
between  the  variables,  they  can  not  each  assume  any  values  at 
random.  There  is  a  fixed  laio  according  to  which  they  must 
change.      This  laio  is  expressed  hy  the  equation. 

Thus,  Newton's  law  of  bodies  falling  from  rest  is  expressed  by 
the  equation 

s=^t'\ 

where  a  is  a  constant,  and  s  and  t  variables,  s  representing  the 
number  of  feet  through  which  the  body  will  fall  in  t  seconds. 


VARIATION.     ALGEBRAIC  EXPRESSION  OF  LAW       315 

It  evidently  expresses   the  law  that  the  distance  through  ivhich 
a  body  will  fall  from  rest  varies  directly  as  the  square  of  the  time. 
Again,  Boyle's  law  of  gases  is  expressed  by  the  equation 

VP=k, 

where  Ar  is  a  constant,  and  V  the  volume  of  a  gas  when  under  the 
pressure  P. 

The  equation  evidently  states  that  the  volume  of  a  gas  varies 
inversely  as  the  pressure  upon  it. 

When  a  law  is  known,  the  equation  which  expresses  it  can 
be  written. 

Example  1.  The  distance  through  which  an  object  will  move 
with  uniform  motion  varies  as  the  velocity  and  time  jointly,  and 
it  is  found  by  trial  that  the  object  with  a  velocity  of  20  feet  per 
second  moves  60  feet  in  3  seconds.  Write  the  equation  which 
expresses  the  law  of  motion. 

If  s=  distance,  v=  velocity,  t=  time,  we  have 

s=kvt. 
Since  s=60  when  v=20  and  ^=3, 

60=A;-20-3. 
Hence,  k=l. 

Therefore,  s=vt  expresses  the  law. 


EXERCISE  92. 

Write  the  equations  which  express  tlie  following  physical 
laws. 

1.  The  force  with  which  moving  bodies  having  the  same 
velocity  strike  a  stationary  body  varies  directly  as  the  masses 
of  the  moving  bodies. 

2.  The  attraction  between  two  bodies  varies  directly  as  the 
product  of  their  masses,  and  inversely  as  the  sqmire  of  the 
distance  between  them. 


316 


ALGEBRA 


3.  If  an  object  is  revolved  at  the  end  of  a  string,  the  tension 
exerted  upon  the  string  by  the  object  varies  directly  as  the 
product  of  the  mass  of  the  object  and  the  square  of  its  velo- 
city, and  inversely  as  the  length  of  the  string. 

4.  The  time  of  vibration  of  a  pendulum  varies  directly  as 
the  square  root  of  its  length. 

5.  The  downward  pressure  upon  the  bottom  of  a  vessel  con- 
taining a  liquid  of  given  density 
varies  jointly  as  the  depth  of 
the  liquid  and  the  area  of  the 
bottom  of  the  vessel. 


, 

Y 

, 

\ 

/ 

\ 

/ 

\ 

/ 

^ 

/ 

y 

/ 

\ 

/ 

\ 

/ 

A. 

V. 

y 

0 

V 

Fig.  14. 


6.  The    amount   of   heat    re- 

-  ceived  by  a  body  from  %  radiat- 
ing source  varies  inversely  as 
the  square   of   the   distance   of 

-  the  body  from  the  source  of 
heat. 

7.  The   resistance  offered  by 
a  wire  to  a  current  of  electri- 

-  city     varies     directly     as    the 
X'  length  of  the  wire,  and  inverse- 
ly as  its  cross-sectional  area. 

214.  As  another  illustration 

-  of  the  expression  of  law  by  an 
equation,  let  us  consider  the 
graph  of  an  equation  with  two 
unknown  numbers. 


The  graph  of  the  equation  y=x^  is  shown  in  Fig.  14.  Here  x 
and  y  represent  the  coordinates  of  any  point  P  on  the  graph. 
If  we  consider  x  and  y  as  variables,  by  changing  their  values 
the  point  P  can  be  made  to  move.     As  the  absolute  value  of 


VARIATION.     ALGEBRAIC  EXPRESSION  OF  LAW      317 


X  increases,  the  point  moves  away  from  the  line  FP,  and  as  the 
absolute  value  of  y  increases  the  point  moves  away  from  the  line 
XX'.  By  making  x  and  y  assume  as  values  all  possible  solutions 
of  the  equation,  the  point  Pcan  be  made  to  take  up  in  succession 
all  positions  on  the  curve. 

Hence,  the  graph  may  he  considered  as  the  path  of  a  point 
moving  subject  to  a  law.  And  the  equation  between  the  coordinates 
expresses  the  law  governing  the  m,otion  of  the  point.  The  point 
must  so  move  that  its  coordinates  continually  satisfy  the  equation. 

This  principle  is  of  great  use  in  mathematical  sciences  such 
as  astronomy  and  physics. 

Example  1.  A  body  is  ||^trn  horizontally  from  a  high  point, 
at  a  velocity  of  16  ft.  per  second.  Find  the  path  of  its  fall  to  the 
ground. 

Let  the  distance  which  it  moves  horizontally  in  t  seconds  be 
represented  by  a?,  and  the  distance  which  it  falls  vertically  in  the 
same  time  be  represented  by  y. 

Then,  x=im. 

By  Newton's  law  of  falling  bodies,  y=lW^. 
Eliminating  t  between  these  two  equations,  x^=16^. 

Measuring  posi- 
tive ordinates 
downward,  and 
positive  abscissas 
to  the  right,  the 
graph  of  this 
equation,  which 
is  the  path  of  the 
moving  object,  is 
found  to  be  the 
portion  of  the 
parabola  shown  in 
Fig.  15. 


oJ 



1 

I  + 

^^ 

\ 

\^ 

N 

\ 

+ 

\. 

1 

V 

Fig.  15. 


318 


ALGEBRA 


Example  2.  A  body  is  thrown  from  the  ground  at  an  angle  of 
45°,  and  with  a  force  such  that  its  horizontal  velocity  is  16  ft. 
per  second.     Find  the  point  at  which  it  strikes  the  ground. 

Let  X  represent  the  distance  the  object  will  move  horizontally 
in  t  seconds,  and  y  the  vertical  distance  it  will  move  in  the  same 
time. 
Then  x=lQt. 

The  vertical  distance  would  be  the  same  as  the  horizontal  dis- 
tance if  the  force  of  gravity  did  not  make  the  body  tend  to  fall 
the  distance  16^^     Hence, 

y=16t-iet\ 
Eliminating  f,  x^=lGx—lQy. 

The  graph  of  this  equa- 
tion will  be  found  to  be 

-  the  portion  of  a  parabola, 

-  shown  in  Fig.  16.  Hence, 
this  curve  is  the  path  of 
the  object's  flight. 

^       Now  the  distance  from 


f 

^ 

■ — 

.... 

> 

y 

N 

v 

/ 

\ 

^ 

— 

— 

— 

— 

^ 

^  the  point  of  ascent  to  the 
Fig.  16.  required  point  is  the  dis- 

tance AB  of  the  figure. 
At  the  point  5,  2/=0-  Hence,  the  distance  is  the  value  of  x  ob- 
tained by  making  2/=0  in  the  equation 

x'=16^-16i/. 
When  2/=0,  x^=z\^x. 

Hence,  a?=0  or  16, 

Therefore,  the  body  strikes  the  ground  16  feet  from  the  point 
of  ascent. 


215.  The  limit  of  a  variable.  If  the  value  of  a  variable 
changes  according  to  some  law,  such  that  the  difference  be- 
tween the  variable  and  a  constant  may  become  and  remain  less 
than  any  assigned  small  value,  which  may  be  taken  as  small 
as  one  wishes,  the  constant  is  called  the  limit  of  the  variable. 


VARIATION.     ALGEBRAIC  EXPRESSION  OF  LAW      319 

Suppose  a  point  to  move  from  X  toward  F,  moving  |  the  distance 
X  Y 

XY  the  first  second,  arriving  at  A  ;  then  ^  the  remaining  dis- 
tance AY  the  next  second,  arriving  at  B  ;  then  }^  the  remaining 
distance  5F  the  third  second,  arriving  at  O  ;  and  so  on  indefi- 
nitely. In  an  infinite  number  of  seconds,  the  point  may  be 
brought  as  near  as  we  please  to  F,  since  we  may  continue  the 
process  of  halving  the  distance  from  the  point  to  Fas  long  as  we 
choose.  Evidently  the  distance  from  X  to  the  moving  point 
gradually  increases^  and  approaches  as  near  as  we  please  to  the 
distance  XY.  Also  the  distance  from  the  point  to  F  gradually 
decreases  and  can  be  made  less  than  any  fixed  small  distance, 
which  can  be  made  as  short  as  we  please. 

Hence,  the  variable  distance  through  which  the  point  has  moved 
from  X  approaches  the  whole  distance  XY  as  a  limit ;  and  their 
difference^  the  distance  from  the  point  to  F,  approaches  zero  as  a 
limit. 

The  sign,  = ,  placed  between  a  variable  and  a  constant,  indi- 
cates that  the  variable  approaches  the  constant  as  its  limit. 
Thus,  a?  ^«  is  read  "  ic  approaches  a  as  its  limit "  and  means 
that,  as  X  changes  according  to  some  law,  the  difference  be- 
tween X  and  a  can  become  and  remain  less  than  any  small 
value  whatever  that  may  be  assigned. 

216.  It  follows  from  the  definition  of  the  limit  of  a  vari- 
able, §  215,  that  if  the  limit  of  x  is  a,  the  limit  of  x — a  is  zero., 
i.  e.,  the  difference  between  a  variable  and  its  limit  is  another 
variable  whose  Vimit  is  zero.     That  is, 

If  x=a,  then  x—a  =  x',  where  x'=0. 
Conversely,  if  x—a  =  x\  or  jc=a  +  jr',  vjhen  x'  is  a  variable 
whose  limit  is  zero  and  a   is   a  constant^  the  limit  of  x  is  a. 
That  is. 

If  x=a  +  x',  where  jr'=0,  then  x=a. 


320  ALGEBRA 

217.  Finite,  infinitesimal  and  infinite  num'bers. 
A  variable  whose  limit  is  zero  is  an  infinitesimal. 

Thus,  if  the  value  of  a  variable  can  be  made  to  decrease  indefi- 
nitely, and  become  and  remain  less  than  any  assigned  value, 
which  may  be  taken  as  small  as  we  please,  it  is  an  infinitesimal. 
A  number,  however  small,  which  does  not  approach  zero  as  a 
limit  is  not  an  infinitesimal. 

If  the  value  of  a  variable  can  be  made  to  increase  indefi- 
mtely  according  to  some  law,  and  become  greater  than  any 
assigned  (or  fixed)  number,  which  may  be  taken  as  great  as 
we  please,  it  is  called  an  infinite  number,  or  infinity.  An  infi- 
nite number  is  represented  by  the  symbol  go  . 

An  infinite  number  does  not  approach  a  limit,  hut  increases 
without  limit.  The  statement  that  x  increases  without  limit  is 
expressed  by  a?=  go  . 

Any  number  which  is  neither  an  infinitesimal  nor  an  infinite 
number  is  called  a  finite  number. 

All^ice^or  definite  numbers  and  all  numbers  heretofore  con- 
sidered ?iYe  finite  numbers. 

218.  If.)  in  a  fraction  inhose  numerator  is  any  finite  num,- 
ber^  except  0^  the  denominator  increases  without  Ibnit^  then  the 
fraction  approaches  0  as  a  limit  /  that  is, 

if  jif=oo  ,  then  ~=0. 

X 

For,  evidently  as  x  increases,  the  fraction  -  decreases  in 
value,  and,  by  making  x  sufficiently  great,  -  can  be  made  less 

X 

than  any  assigned  value,  which  may  be  taken  as  small  as  we 
please. 

Thus,  y\,  T^o,  iwo,  TO  loo,  TTTo^offo-)  ©tc. ,  are  a  series  of  fractions 
in  which  each  fraction  is  one-tenth  the  preceding  one,  and 
each  denominator  is  ten  times  as  large  as  the  preceding  one. 


VARIATION.     ALGEBRAIC  EXPRESSION  OF  LAW       321 

If  this  series  of  fractions  is  carried  out  indefinitely,  a  fraction 
will  be  reached  whose  value  is  less  than  any  assigned  value, 
however  small. 

219.  Properties  of  zero.     Zero  may  be  defined  as  the  differ- 
ence between  two  equal  numbers  ;  that  is, 

0=n—n. 

Thus  defined,  it  leads  to  some  interesting  results. 

(1)  Let  a  be  any  finite  number.     Then 

a(i  =  a{n—ri) 
=^an—an 
=  0. 
Hence,  the  product  obtained  by  inidtiplying  any  finite  number 
by  zero^  equals  zero. 

(2)  If  a  be  any  finite  immber,  then 

^  _n  —  n 
a        a 

n     n 

~^  a     a 

=  0. 

Hence,  the  quotient  of  zero  divided  by  any  finite  number^  equals 
zero. 

(3)  The  symbol  ^  represents  a  number  which  multiplied  by  0 

gives  0. 

But,  according  to   (1),  any  finite  immber  multiplied  by  0 
gives  0. 

Hence,  ^  represents  any  finite  number  ichatever. 
The  fraction  ^  is  called  an  indeterminate  fraction.* 


*  In  like  manner  it  can  be  shown  that  the  symbol  §  is  indeterminate. 


322  ALGEBRA 

(4)  The  fraction    -  where  a  is  any  finite  number^  represents 

a  number  which  multiplied  by  0  gives  a. 

But,  according  to   (1),  no  finite  number  multiplied   by  0 
gives  a. 

Therefore,  tt  has  no  finite  value  lohatever. 

If  a  is  finite  and  a?  is  a  variable  whose  limit  is  zero,  then  by 

taking  £c  smaller  and  smaller,  -  can  be  made  to  exceed  any  fixed 

number  which  may  be  chosen  as  great  as  we  please.     Hence, 

the  limit  of  -    when  cc  =  0,  cannot  be  found;  that  is,  it  exceeds 

all  definite  fixed  numbers^  or  is  infinite.     This  is  expressed  by 

a. 

Mut  this  is  not  an  equation  in  the  ordinary  sense. 

(5)  Since  ^  is  indeterminate,  and  ^y  is  not  finite,  therefore, 
it  clearly  is  not  allowable  to  divide  by  0.     See  foot-note  to  §  24. 

Let  the  student  show  the  fallacy  in  the  following  reasoning. 

Let  a=h. 

Multiplying  by  a,  a^=ab. 

Subtracting  6^,  a^—b'^=ab—¥. 

Factoring,  {a-\-h){a—b)=h{a—h). 

Therefore,  2b{a—b)=b{a—b). 

Divide  by  6(a-6),  2=1. 

220.  Indeterminate  fractions.      For   certain  values  of  the 

variable,   a  fraction  will   sometimes   take  the    indeterminate 

.        0 
form  -^ 


VARIATION.     ALGEBRAIC  EXPRESSION  OF  LAW       323 
Thus,  when  a  =  2,     _i)  =  m   ^^  indeterminate  form.     This 

fraction,  however,  is  equal  to  ^^ _,^ Now,  as  long  as 

a   is   7iot  equal   to  2,  we   may  divide   both  terms   by  a— 2, 

hence  -^^ ^^^ -  =  a  +  2,  when  a  is  not  equal  to  2. 

If  a=2,  by  the  definition  of  a  limit,  a  cannot  become  2, 
although  it  differs  from  it  ever  so  little  ;  hence  we  may  divide 
by  a— 2.     Now,  as  a  approaches  2,  evidently  a  +  2  approaches 

(f 4 

2  +  2  or  4.     Hence  we  say ^=4  as  a=2.     Hence, 

•^  a  —2 

a'^—4. 

4  is  called  the  value  of  -.  when  a=2. 

a  —2 

The  value  of  such  a  fraction,  for  any  particular  value  of  its 
variable,  is  defined  as  the  limit  which  the  fraction  approaches 
when  the  variable  approaches  this  particular  value  as  its  limit. 

Example.     Find  the  value  of  — r-— -  when  x=a. 

oc — a 

When£c=a  we  have  =-,  the  indeterminate  form. 

x—a     0 

For  any  value  of  x  other  than  a,  we  have 

x—a 
Now,  as  X  approaches  a  this  expression  evidently  approaches 

n^^ O'i 

a^  +  aa  +  a^  or  3a^,  the  value  of  -— — —  when  0?= a. 

X  ^~a 


CHAPTER  XXI. 

FRACTIONAL  AND   NEGATIVE   EXPONENTS. 

221.  The  following  five  fundamental  laws  of  exponents 
have  been  established  in  Chapter  V,  Chapter  VI,  and  Chapter 
VII: 

(A)  «"'a"  =  a'^+"; 

(C)  («'")"  =  «'»"; 

(D)  (aby  =  a''b''l 


(-)  (!)"=? 


In  all  the  work  heretofore,  the  exponent  has  been  a  positive 
integer.  Where  the  exponent  is  a  positive  integer,  by  defini- 
tion it  indicates  the  number  of  times  that  the  base  is  to  be 
used  as  a  factor.  Thus,  d^  means  a- a- a.  The  proofs  of  these 
five  laws  were  based  upon  the  definition  of  an  exponent.  And 
therefore  these  laws  were  established  only  when  the  exponents 
were  positive 


22t2t.  Fractional  and  negative  exponents.  We  have  seen 
that  by  extending  the  operations  with  numbers  so  as  to  make 
them  general,  new  kinds  of  numbers  have  been  conceived ; 
viz.,  fractional,  negative,  and  imaginary  numbers.  Likewise  in 
the  attempt  to  extract  a  root  of  a  power  of  a  number  by  the  law, 

a  fractional  exponent,  a  neAV  kind  of  exponent,  is  sometimes 

824 


FRACTIONAL  AND  NEGATIVE  EXPONENTS  325 

obtained  if  this  law  be  extended  to  hold  for  all  j^ositive  integral 
values  of  m  and  n. 

Thus,  y'd^=a^^'^  or  a^- 

If  the  law  a'"^«" =«"»-"  be  extended  to  hold  when  n  is  greater 
than  m,  then  a  negative  exponent  will  result. 

Thus,  a^-^-a5=a=*"^=a"2. 

It  thus  appears  that  the  extension  of  one  process  gives  rise 
to  fractional  exponents,  and  the  extension  of  another  to  negative 
exponents. 

It  is  clear  that  a  fractional  or  negative  exponent  cannot  he 
given  the  same  meaning  as  an  exponent  which  is  a  j^ositive 
integer. 

3 

For  example,  a^  cannot  indicate  that  a  is  to  be  used  as  a  factor 
I  of  a  time.  Such  a  statement  is  absurd.  Also,  a-*  cannot  in- 
dicate that  a  is  to  be  used  as  a  factor  —4  times. 

We  shall  define  operations  with  negative  and  fractional  ex- 
ponents so  that  fundamental  law  A,  §  221,  shall  be  true  also 
Avhen  m  and  n  are  fractional  or  negative.  We  may  then  in- 
vestigate the  meanings  for  such  exponents  as  are  required  by 
this  law. 

223.  Meaning  of  a  positive  fractional  exponent. 

Since  fractional  exponents  are  to  be  so  defined,  that  law  A 
shall  be  true  for  such  exponents,  then  if  m  and  n  be  any  positive 
integers. 


m  ,  m  .  m 


•     to  n  terms 


a".a".a"  .  .  .  to  ^i  factors  =«"  "  "  (A), 


m 


or. 


Hence,  a*'~|/a'",  by  the  definition  of  a  root, 

=  (f'ar.  §  147. 


326  ALGEBRA 

Therefore,  a  positive  fractional  exponeiit  indicates  a  root  of 
a  power ^  or  a  poioer  of  a  root^  of  its  base  ;  the  numerator  indi- 
cating the  power ^  and  the  denominator  indicating  the  root. 

Thus,   a"^=^i^a^,  or  (i^a)^  a^=i^^  16^= (1^16)3= 2^ =8. 

It  follows  that  in  any  expression,  any  indicated  roots  may  be 
changed  to  fractional  powers,  and  vice  versa. 

^  g  .___  3/"~"  -2  11  2 

Thus,  y  x'^  +  y  xy  +  y  y^  meiy  be  written  x^^+x^^'y^  +  y^. 

Note. — Keducing  a  fractional  exponent  to  higher  or  lower 
terms  does  not  change  the  value  of  the  expression. 

TO 

For,  a"  =  V^ 

=17  «^^  §  148. 

mx 

Hence,  fractional  exponents  that  may  occur  in  using  laio  A 
tnay  he  reduced  to  a  common  denominator^  then  added. 

224.  Meaning  of  a  negative  exponent.  Since  a  negative  ex- 
ponent is  to  be  so  defined  that  law  A  holds,  if  n  be  any  positive 
integer  or  a  fraction,  then,, 

a"  •«""  =  «"""  Law  A 


48. 


Dividing  both  members  by  w\   «""=—-. 

a 


Hence,  a  negative  exponent  indicates  a  power.,  or  a  power  of  a 
root,  of  its  base  that  is  to  be  used  as  a  divisor. 

Thus,  a-^=\  or  in-a*;  a^lr^  ^  —  ova^-^lf;  a~^=~^-^. 
«  '6^  at    pa' 


FRACTIONAL  AND  NEGATIVE  EXPONENTS  327 

225.  Atiy  factor  may  he  changed  from  the  dividend  to  the 
divisor^  or  from  the  divisor  to  the  dividend,  by  changing  the 
sign  of  its  exponent. 

This  follows  from  the  interpretation  of  a  negative  exponent. 


Example  1. 


7y^z-.^       7a' 


Example  2.       ^i^=^ip  or  a-^h-x-y-^,ov  ^,^-,^,y,, 

_6        1  1  11 

Examples.  4  ^=— = — ^~— 17^—7:7^' 

4I     (^4)5     25     32 


EXERCISE  93. 


Express  as  roots  of  integral  powers  ; 

4                                                            2  2 

1.  a'T.                           4.  b^.  7.  x'^. 

i                                       i_o  5 

2.  a2.                            5.  ^7  .  8.  83. 

3.  x^,                            6.  yi  9.  27i 

10.  it\i. 


Write  with  positive  exponents : 

11.  a;2y-3.                    ^^    lar^b-^c  ^^    2x(x'^-y^)-^ 

12    a-i^>2c-2                    '  ^xy-'^z-'^'  '     Sy(x-y)-''  ' 

*^^     ^  (a— ^)(a4-^)^ 

14.    3x-5y22-3.                              ^_1 05-4^-1  2     _3 

ly. -. — 5 — .  ^,    x^y  '^ 

2                                       xy-^z-^  24.         ^ 

15  -r27/-is-2                           ^  ^-1 

16  5^'                        21    (^^-^)"'  25    <^'"^'>"^. 


328 

ALGEBRA 

Write  as 

roots  of  positive  integral  powers. 

26.  a~t 

2 

29. 

■■  2  ■* 

31. 

1 

i* 

27.  x~^. 

^ 

2/  « 

28.  A. 

C 

30. 

33.  (^^-^)"f. 

32. 

2(a  +  5)-t 

Find  the  value  of 

: 

34.  di 

37. 

625"* 

40. 

(-64)-*. 

35.  si 

38. 

(»)-'■ 

41. 

125~*X3^ 

36.  27"  i 

39. 

©-'■ 

42. 

8*x9-t. 

43 

25"^ 
•  81""^ 

"  & 

/1\| 

226.  Having  defined  operations  \oith  fractional  and  negative 
exponents  as  being  required  to  satisfy  the  laic  a'"-a"  =  a'"^",  toe 
loill  71010  shoio  that  they  icill  satisfy  the  other  latos  of  exponents^ 
also  :  viz  :  .^'''  .     '■' 

Law  B^  ar-T-a"-  a'""". 

Law  (7,  (a'"y  =  a'"". 

Lawi>,  (aby  =  a"b". 


Law  JS; 

(» -F 

I.  When  the  exponents 

are  fractional. 

(1)                                  a-n-^. 

-a<i  =  a''-a   «. 

§225. 

§  222. 


FRACTIONAL  AND  NEGATIVE  EXPONENTS  329 
Hence,  law  B  is  satisfied. 

(2)                                   («T.)^=(^   ^^)  §223. 

^Cx^W^Y  §  149. 

=  \/Wy  §  148. 


=«««,  or  a^^'  1.  §  223. 

Hence,  law  C  is  satisfied. 

(3)  {ab)-^=^y{ahY  §  223. 


=  l/cr^>'»  X>,  §  221. 

=i;^^i/ F  §  67. 

Hence,  law  D  is  satisfied. 

=K^  js;§22i. 

=^r^  §  68. 

y  b'" 

m 

Hence,  law  J^  is  satisfied. 

II.  When  the  exponents  are  negative. 

(1)  6«-"*^a-"=a-'"a'\  §  225. 

=«-'»+".  §  222. 


330  ALGEBRA 

Hence,  law  B  is  satisfied. 

(2)  (^a-)-^=(L}j-^  §225. 

.  =  (<:r)"  §  225. 

=  cr\  or  a'-"'''-'''  E,  §  221. 

Hence,  law  G  is  satisfied. 

(3)  W-"-=(i'^  §225: 


1 
1    1 


§221. 


=  a-''-b-\  §  225. 

Hence,  law  D  is  satisfied. 

(4)  vv    y«y'  §225. 

^* 

=  ^* 
a" 

Hence,  law  E  is  satisfied. 


225. 


227.  The  consideration  of  the  laws  when  the  exponents 
are  surds  or  imaginary  numbers  will  not  be  taken  up  in  this 
book. 

228.  In  the  following  exercise  we  may  make  use  of  the  five 
laws  of  exponents,  which  have  now  been  interpreted  for  nega- 
tive and  fractional  exponents,  as  well  as  for  integral  exponents. 


FRACTIONAL  AND  NEGATIVE  EXPONENTS  331 

All  radical  signs  should  first  he  replaced  by  fractional  expo- 
nents. 

Example  1.     Simplify  a-^- a*. 

2         3  8  9 

1  7 

=a;^^. 

'    Example  2.     Simplify  a?~^-^a7"^". 


-2  _    3  _3_/'_    3\ 

4,3 


■x 


To 


Example  3.    Simplify  (4a"*6~^)~^. 

3' 
4^ 


8 


3/-^       /   I-.   2       _1 


Example  4.    Simplify   y=I:-(^)  -'^• 

_i 


332  ALGEBRA 

3,—  3, 3,— 


Example  5.     Multiply  yx^—Vxy  +  vy^  by  ^/x+^/y. 

3/—  3 3  —  2  1_     1  a 

Vx^—Vxy+Vy^=x^—x^y''^+y^. 


5./73.r/2 


3  3    -J  ^ 


8  It  2 

x^—x^y^  +  y'^ 


x^  +  y^ 


2      1  12 

_?      1  12 

+  a?^  2/      x^y'^  +  y 
X  +y 

EXERCISE   94. 


Simplify : 
1    9^-2.7.-3.^7  _7   3         1      1 

2.  a 


1        .3  16.  6a   2^.3^(3a   */>  i). 

3.  a2.^-2.«2.  ^^   ^^ 

2      1  16.  («   2^  V^- 


4.  5iK    a-3ie2.  _2      1        3 

4  1  17.  (4.^   3y    2^)-^^. 

6.  a-2-^a-4.  18.  {-^la-^x   4)    3. 

7.  x-i^x"^.  19.  (32^-tyH^)"i 


8.  a2^-3_^f^-i.7.-i,  20.  i/av'aae. 

9.  a-lJ-2c-3-^«3^;2e-4.  21.    T^-P^-^1>^. 

_4  _7.  5 ,  5 

10,  711    ^—771    5.  22.  i/a-2a;-i---va-i  23.-6. 

1^-  y'-^y-'-  23.  i>v^-^-2. 

12.  5tV^(^-i^.-A).  24,  |/^3^^^2. 

13.  aHic~^--(aZ»"3c"i).  26.  y  x^i/^x^y'x^^. 


FRACTIONAL  AND  NEGATIVE  EXPONENTS  333 


o/j    V^^        ya^b^  30.  Qi/a-^---2V  cr^. 

26.   "i/—^-^  12 


Va^b     Va^b^  „^  _i 


^  31.     (0^2  _y2)      2.^/(^_^.y)3. 


28.  i^a/«V«^)^  33    ^/^JaW 

29.  a"(-«-'^).  •    I      ^-3  •  \^,-i^ 

4       2.2.       44       22       4 

34.  {x^^-x^y^^y^){x^—^y^-^y^). 

35.  (a^— a^  +  a^-a~i)(a^  +  a^). 

36.  {x^ ^x^y^ -\-y^){^ —y^). 

37.  (2aj-i-3cc-7)(5  +  4i«-i). 

38.  (i>a4  +  T/a2i>^2  4_-^^4-)(^^^^2_^V^2). 

39.  {Vx'^  —  \^'xy^\/y''--){Vx^ yy). 

/    4  6  \   /    2  \ 

40.  (-X^— 3^  +  9)(-x:r-+3]. 
\l/a;4     yx'^       JWx'^       J 

41.  (a  +  2i^a2_3^>--)(2-4a~i-6a~4). 

42.  {x^+y^)^(x^  +  yi), 

44.  (x'^—2  +  x~^)~{x-^—x~'^). 

45.  (^>5  +  2^»-5  +  7)(5-3/>»-5+2^>5). 

46.  (i/^  +  v'^+l)(£c~^+a;~i  +  l). 

47.  (8a-2-8a2+5a6_3^-6)^(5^2_3^^-2)^ 

48.  (a^  +  Z>~*)2.  50.  (x^  +  xi  +  l  +  x~~^-{'X~^)^-. 

49.  (a;~i-2/^)2.  51.  (ai  +  ic~^)(a^-a;~4). 


334  ALGEBRA 

62.  (3£c~^-4y^)(3a;~*  +  4y^).      57.  {a'' -\-b^)-^{a^-\-h^), 
53.  {a-4:h)^{ya-'li/b).  68.  {x-y)^{\/~x-vy). 

^  ^     ^  ^  59.  8    3(252-81M. 

55.  (a-2--9)--(a-"-3).  ^  ^ 

66.  (a-4-l)--(a-i-l).  60.  27^(27^+3^). 


EXERCISES  FOR  REVIEW  (VI). 

1.  Define  ratio  ;  terms  of  a  ratio.     Illustrate  each. 

2.  What  are  commensurable  numbers?  Incommensurable 
numbers  ?    Illustrate. 

3.  What  is  a  j^?rop(9r^^o?^?  Illustrate.  Name  the  ^erws  of  a 
proportion. 

4.  Show  that  the  product  of  the  extremes  in  any  proportion 
equals  the  product  of  the  means. 

6.  What  is  a  test  of  the  correctness  of  a  proportion  ? 

6.  Is  2a V  :  3aic^=6aic^  :  Stc^"  a  true  proportion?  Give 
proof. 

7.  Give  five  fundamental  principles  in  proportion,  and  illus- 
trate each. 

8.  Define  mean  proportional ;  third  p>roportional ;  fourth 
proportional. 

9.  Find  the  mean  proportional  between  ^x  and  ^x^ ;  between 
72aj^y  and  ^x^\f\  between  4(a  +  ^)  and  9(a  +  ^)^;  between  a 
and  X. 

10.  Find  the  third  proportional  to  1  and  8 ;  tox*  and  l^x^a^ ; 

to  -l^^y  ^^^  25y^;  to  — t~x  ^'^^  {a-\-by. 

11.  Find  the  fourth  proportional  to  9,  7,  and  3;  to  1,  x^^  and 
2y ;  to  !«,  2aJ,  and  bx ;  to  x—y^  ^^~y^  and  1. 

12.  What  is  a  continued  proportion  ? 


FRACTIONAL  AND  NEGATIVE  EXPONENTS  335 

13.  Show  that  if  -=-==-=-,  then  — , —  =—, — 

X     y     z     w*  x-\-y     z  +  w 

14.  Show  that  if  t  =  ^i^  then  -j=  \  ^ 

b     cV  d    ^c'-^d' 

16.  Define  a  variable.     A  constant.     Illustrate  each. 

16.  What  is  direct  variation? 

17.  If  £c  oc  2/,  and  £c=6  when  y=4,  find  y  Avhen  £c=2. 

18.  If  icoc  y,  and  a;=10,  when  y=l,  write  the  equation  con- 
taining X  and  y. 

19.  If  icocy,   and   when   i»=f,  y=9,  express  the  equation 
between  x  and  y. 

20.  What  is  inverse  variation  ? 

21.  Given  that  x  varies  inversely  as  a^  also  that  when  iK  =  2, 
a =5.     Find  aj  when  a  =  10. 

22.  Given   that  x  varies   directly  as  y  and  inversely  as  z. 
W^hen  jc=3,  y=5,  and  z=l.     Find  2  when  aj=3  and  y=15. 

23.  What  is  meant  by  the  limit  of  a  variable  ? 

24.  What   is  an   infinitesimal?     An    infinite    number?     A 
finite  number  ? 

25.  What  is  the  value  of  -  ?     Of^?     Of^?     Of—? 

iC  0  0  QO 

26.  Express  in  symbols  the  five  fundamental  laws  of  ex- 
ponents.    Prove  each  when  m  and  n  are  positive  integers. 

27.  Simplify  ««•«";  a'^a'-,  {ay-,  {ct'b'f;   (^^' 

2,  m  3 

28.  What  is   the  meaning  ofa^?     Of  «'» ?     Show  that  a^ 
means  ya^.  ^ 

29.  Simplify  (a^)^^'  ;    {x'Y  ;  (2aM)«. 

30.  Find  the  value  of  4* ;  8^ ;  25^ ;    (J^)i 


336  ALGEBRA 

31.  What  is  the  meaning  oia-'^  ?  Of  «-"  ?  Show  how  this 
interpretation  of  a  negative  exponent  is  derived. 

ct  ^        2 1*  (/ —  ^ 

32.  Write  with  positive  exponents -Trr2  ;  q-^'^^zTg  ;  2(a—b)-^. 

33.  Find  the  value  of  4~^;  5-2^5-3;  4~^;  25"i;(-8)t- 

34.  Simplify    (a-^y^ ;    (a'^fi;    (i/^^T^;    (aJ^y-^)-^  ; 

35.  Write   the   value   of    (x'i-^-y-^y;    (2a-^-Sb~^y; 

36.  What  is  an  ineqiiality  ?  Distinguish  between  con- 
ditional  inequalities  and  identical  inequalities.  To  which 
class     does     a^-^b^^^ab    belong?      To     which     class     does 

^3>^|  +  5  belong? 

37.  What  is  meant  by  the  solution  of  an  inequality  ? 

38.  Solve  the  conditional  inequality  in  Exercise  36. 

39.  Given    |  ^""12^=-^^'     ^^"<^^  ^^^  ^"^^^''^  ^*  ^  ^^^^  V- 

40.  A  teacher,  being  asked  the  number  of  his  pupils,  replies 
that  twice  their  number  diminished  by  7  is  greater  than  29, 
and  three  times  their  number  diminished  by  5  is  less  than 
twice  their  number  increased  by  15.  Find  the  number  of 
pupils. 

41.  Three  times  a  certain  number  plus  16  is  greater  than 
twice  that  number  plus  24,  and  |  of  the  number  plus  5  is  less 
than  11.     Find  the* number. 


ia 


—  23 


CHAPTER  XXII. 
PERMUTATIONS  AND  COMBINATIONS. 

229.  In  this  chapter  we  shall  discuss  an  important  class  of 
problems  which  the  following  examples  will  illustrate. 

Example  1.  In  how  many  ways  can  a  program  be  arranged 
consisting  of  a  solo,  a  debate  and  an  oration  ? 

By  putting  any  one  of  the  three  numbers  first  and  each  of  the 
remaining  two  numbers  after  it,  we  get  the  following  six  ar- 
rangements :  (1)  Solo,  debate,  oration ;  (2)  solo,  oration,  debate  ; 
(3)  debate,  solo,  oration  ;  (4)  debate,  oration,  -solo  ;  (5)  oration, 
solo,  debate;  (6)  oration,  debate,  solo. 

Example  2.  How  many  different  numbers  of  two  digits  each 
can  be  formed  from  the  digits  2,  3,  4,  5,  using  each  digit  but 
once  in  the  same  number  ? 

We  can  take  each  with  one  of  the  others  ;  hence,  we  get  the 
following  twelve  numbers: 

23,  24,  25,  32,  34,  35,  42,  43,  45,  52,  53,  54. 

These  examples  illustrate  the  general  problem  of  finding  the 
number  of  arrangements  of  a  certain  number  of  things  taken 
from  a  given  number  of  things. 

230.  Permutations.  All  the  possible  arrangements  that  can 
be  formed  from  the  different  groups  of  ..r  things  which  can  be 
taken  froni  n  different  things,  are  called  the  permutations  of 
n  things  taken  r  at  a  time. 

Thus,  the  permutations  of  the  three  letters  a,  &,  c,  taken  one  at 
a  time,  are  a,  6,  c     Taken  tn^o  at  a  time,  they  are  a6,  6a,  ac^ 
22  337 


338  ALGEBRA 

ca,  he,  cb.     Taken  three  at  a  time,  they  are  abc,  acb,  bac,  bca, 
cab,  cba. 

It  is  clear  that  two  permutations  are  different  unless  they  con- 
tain the  same  things  arranged  in  the  same  order.  Thus,  ab  and 
ba  are  different  permutations. 

The  number  of  permutations  of  n  distinct  things  taken  r  at 
a  time  is  represented  by  the  symbol  ^Z'^. 

Thus,  3P2  represents  the  number  of  permutations  of  3  things 
taken  two  at  a  time. 

4P2  represents  the  number  of  permutations  of  4  things  taken 
2  at  a  time. 

jf^Pg  represents  the  number  of  permutations  of  10  things  taken 
6  at  a  time. 

231.  The  value  of  ^Z',.,  From  the  illustrations  in  §  230  we 
have  seen  that  ^P^  =  S,  ^P^—^^  3^3=6.  There  is  a  law  by 
which  the  number  of  permutations  in  any  case  may  be  writ- 
ten. The  law  may  be  derived  by  the  use  of  the  following 
principle : 

Jff"  a  thing  can  be  done  in  a  dijlfere7it  tcai/s,  and  a  second 
thing  can  he  done  in  b  different  ways  without  interfering  with 
the  firsts  there  vnll  he  ab  wags  of  doing  the  tico  things. 

The  truth  of  this  principle  is  evident. 

To  illustrate  it,  suppose  that  5  boats  are  plying  between  two 
cities.  Find  the  number  of  ways  in  which  a  person  may  go  and 
return  from  one  city  to  the  other  by  a  different  boat. 

Evidently,  in  going,  he  has  the  choice  of  any  one  of  the  5 
boats.  In  returning  he  has  a  choice  of  any  one  of  the  4  not  used 
in  going;  hence,  with  any  one  of  the  five  choices  he  has  four 
others,  or  5  x  4  in  all. 

The  number  of  ways  that  n  things  can  be  taken  from  n 
things  one  at  a  time  is  evidently  n.    Hence, 

nP.=n.  (1) 


PERMUTATIONS  AND  COMBINATIONS  339 

From  9\  things  one  thing  can  be  taken  in  n  different  ways, 
and  after  tliis  is  done  a  second  thing  can  be  taken  from  the 
remaining  n—1  things  in  71  — 1  different  ways.  Hence,  from 
the  preceding  principle,  there  are  ?i(/i—l)  ways  of  taking  the 
two  things.     That  is, 

^P,=?i(n-1).  (2) 

Thus,  5P2=5-4=20;  i2P2=1211=132. 

After  the  first  two  tilings  have  been  taken  in  any  one  of  the 
n(n—l)  ways,  the  third  thing  can  be  taken  from  the  remain- 
ing ?i—2  things  in.  n—2  ways.  Hence,  there  are  n{?i—l)(n—2) 
ways  of  taking  the  three  things.     That  is, 

,,P,=n(n-l){n-2).  (3) 

Thus,  4P3=4-3-2=24.  ^,P^=10-9-8=720. 
In  like  manner, 

,P,=n(n-l){n-2)(n-8)(n-4.);    ^'  (5) 

,P,=n(n-l)(?i-2)(7i-S)(n~4:)(n-b) ;  (6) 

and  so  on.  ^5  4        ^  X        / 

Kow  from  (1),  (2),  (3),  (4),  (5),  and  (6),  Ave  see  that  ^P,  has 
1  factor ;  ^^P^  has  2  factors ;  ,,P^  has  3  factors ;  ^P^  has  4  fac- 
tors ;  jjPg  has  5  factors  ;  ^Pg  has  6  factors  ;  and  so  on.  And  in 
general  „P^  will  have  r  factors,  the  last  one  being  7i—(r—l), 
or  n  —  r-\-l.     That  is,  ^  <>-    <ri.l^    ■: 

JP,=h{n-\){n-%)  •  •  •  •  •  (/i-r+l).  (7) 

Thus,  8P5=:8-7-6-5-4=6720. 

If  n  things  are  taken  all  at  a  time,  then  r=w.  Hence, 
from  (7), 

„/>„  =  /7(/7-l)(/7-2) 4  3  21.  (8) 


340  ALGEBRA 

232.  Factorial-/?.     In  (8)  of  §  231,  the  product    . 

?i(7i-l){7i-2) 4-3-21 

is  called  factorial—/?,  and  is  usually  designated  by  the  symbol 
I  )i,  or  n\    . 

Thus,  |^  =  6-5-4-3-21=720;  5!  =  5-4-3-21  =  120. 

Hence,  from  §  231,  we  have 

Example.  In  how  many  ways  can  5  books  be  arranged  on  a 
shelf  ? 

The  number  =,P,,  or  |_5=5  4-3-21=120. 

233.  When  the  things  permuted  are  not  all  different. 

In  many  problems  the  things  permuted  are  not  all  different. 
We  sliall  now  determine  the  number  of  permutations  in  such 
cases. 

Suppose  that  a  of  the  n  things  are  alike,  and  suppose  that 
we  form  the  iV^  permutations  of  the  ??-  things  taken  ?i  at  a  time. 
Now,  if  in  any  one  of  these  permutations  the  a  like  things  be 
replaced  by  a  unlike  things,  different  from  all  the  rest,  then  by 
changing  the  order  of  these  a  new  things  only,  we  can  form 
I  a  new  permutations  from  the  one  permutation.  This  can  be 
done  in  the  case  of  each  of  the  JV  permutations.  Ileuce,  in 
all,  ]}^\a  new  permutations  can  be  found.     Therefore, 

JSr\a  =  „P,,  (all  different), 
and  N  =  '^=^. 

Similarly,  it  can  be  shown  that,  if  a  of  the  n  things  are  alike, 
and  h  others  alike,  then 


PERMUTATIONS  AND  COMBINATIONS  341 

And  if  a  of  the  n  things  are  alike,  h  others  alike,  and  c  others 
alike,  then 

\n 


^      \a     \h     \c    ' 


and  so  on. 


Example  1.  How  many  different  numbers  can  be  formed  by- 
using  all  of  the  figures  2,  2,  2,  3,  3,  4,  5,  5,  5,  5  ? 

110 
The  number  =        -•        =12600. 

li  It    li 

Example  2.     Find  the  number  of  permutations  of  the  letters 
of  the  word  Indiana. 

Here  there  are  two  e's,  two  n'fe,  two  a's,  one  d.     Hence  the 
number  =j^   g  ^^  =630. 


EXERCISE  96. 

Find  the  value  of : 

1-  li-  L^  li  12.  ,p,.        ' 

2.  16.  '•       ^    ■  ''■  '^^-  .    , 

|2    13    14  14.  1.A-  V' 

3.  |3|5.  8.^^-  15.  „P,. 

4.  |4    |2.  [6    |8  16-  i«^«- 


17 


9.  -n^-  17.  ,,A. 

'•  T  14    15    16  ''•  'J- 

[8_         ^°-  Tir'    19-  # 


342 

ALGEBRA 

Show  that 

21.  n(n-l){n-2)  \n-d 

\n-l 
22.    1 --\n-2. 

y^ 


!y('^^y^^X 


n-l 


23.    \a  •  \a  •  (a  +  l)~a  =  \a  +  l  ■  \a  —  l. 

24.  How  many  numbers  of  three  digits  can  be  formed  by 
using  the  digits  1,  2,  3,  4,  5,  using  each  digit  but  once  in  tlie 
same  number  ? 

25.  How  many  numbers  of  two  digits  can  be  formed  bj^  using 
the  digits  2,  4,  6,  8  ? 

26.  How -many  permutations  can  be  formed  of  the  letters  of 
the  alphabet  taken  twji-at  a  time  ? 

27.  In  how  many  different  ways  can  5  boys  stand  in  a  row  ? 

28.  If  8  steamers  ply  between  Liverpool  and  New  York,  in 
how  many  ways  can  I  go  by  steamer  from  New  York  to  Liver- 
pool and  return  by  a  different  steamer  ? 

29.  Six  ladies  and  six  gentlemen  are  to  be  seated  about  a 
circular  table.  In  how  many  different  positions  can  they  be 
seated  so  that  there  shall  be  a  gentleman  at  the  right  of  each 
lady? 

30.  In  how  many  ways  can  a  class  of  15  pupils  be  seated  in 
15  seats? 

31.  In  how  many  ways  can  2  different  prizes  be  awarded  to 
10  boys  so  that  no  one  boy  gets  both  prizes? 

32.  How  many  permutations  can  be  made  of  the  letters  of 
the  word  A?ina  ?     The  word  Missouri  ? 

234.  Combinations.  The  different  groups  of  r  things  that 
can  be  taken  from  n  things,  when  the  arrangement  is  not  jcmi- 
sidered,  are  called  the  combinations  of  the  ?^  things  taken  r  at  a 
time. 


PERMUTATIONS  AND  COMBINATIONS  343 

Thus,  the  combinations  of  the  letters  a,  6,  c,  d^  taken  two  at  a 
time,  are  ah,  ac,  ad,  be,  bd,  cd.  Taken  3  at  a  time,  they  are 
abc,  abd,  acd,  bed. 

It  is  clear  that  two  different  combinations  can  not  contain 
the  same  things  arranged  in  different  orders. 
Thus,  abc  and  acb  are  the  same  combination. 

235.  The  number  of  combinations  of  n  things  taken  r  at  a 
time  is  represented  by  the  symbol  „Cr. 

Thus,  4C3  represents  the  number  of  combinations  of  4  things 
taken  3  at  a  time. 

236.  The  value  of  „^^,  The  number  of  combinations  of  n 
things  taken  r  at  a  time  is  easily  found  by  establishing  the 
relation  between  „  C^  and  „P^. 

Suppose  n  different  things  combined  r  at  a  time.  .  Every 
combination  of  the  r  different  things  will  have  |r  permutations, 
taken  r  at  a  time.  Hence,  the  total  number  of  permutations 
will  be  „CV  Ir  ;  that  is, 


Therefore, 
Thus, 

237.  If,  in  the  value  for  ^^C\  found  in   §  236,  we  replace 
„P,.  by  its  value  found  in  §  231,  we  get 

_n{n-1)(n-2)  ■ (/i-r+/) 

nC/^  It;;  •  v-"-; 

It  is  sometimes  useful  to  express  the  value  of  „  C^  in  a  dif- 
ferent form. 


nCr 

L=nP. 

,Cr  = 

C?a  = 

5-4-3 
~3-21 

10. 

344  ALGEBRA 

Multiplying  the  numerator  and  denominator  of  the  fraction 
in  (1)  by  \?i—r,  we  get 

7i(n—l){n—2) (n—r-i-l)\?i—r 


r     \n—r 


or  ^C=r-^=—  (2) 

r  \n—7' 


n^  r 


If  we  replace  rhjn—r  in  (2),  we  get 

\n  In 

C     = ^= =_!=___.  (3) 

^    ""''     \7i—r   \7i—n  +  r     \n—r    \r 

From  (2)  and  (3)  it  follows  that 

nCr  =  nCn-r'  (4) 

Thus,  50C48  =  50C2=^  =  1225. 


EXERCISE  96. 

Find  the  value  of  : 

1.  ^Cy               3.  80 ^n- 

5.      12  Cg.                          7.     gOg. 

9'  25620- 

2.   joCg.           4.  15612. 

6.  8  63.             8.  206^3. 

10.  ,C',. 

11.  Find  the  number  of  combinations  of  12  things  taken  2 
at  a  time  ;  taken  3  at  a  time  ;  taken  9  at  a  time. 

12.  How  many   selections  of   3   books  can  I  make  from  5 
books  ? 

13.  How  many  combinations  can  be  made  from  the  letters 
a,  b,  c,  d,  e,  taken  three  at  a  time  ? 

14.  How  many  combinations  can  be  made  from  the  26  let- 
ters of  the  alphabet,  taken  2  at  a  time  ? 

15.  How  many .  different  products  can  be  formed  from  the 
numbers  2,  4,  6,  8,  if  each  product  contains  3  unequal  factors  ? 


PERMUTATIONS  AND  COMBINATIONS  345 

16.  In  how  many  ways  can  a  committee  of  3  be  selected 
from  8  men  ? 

17.  How  many  different  committees  can  be  formed  from  10 
Republicans  and  6  Democrats,  if  there  are  2  Republicans  and 
1  Democrat  on  each  committee  ? 

18.  There  are  8  points  in  a  plane,  no  three  of  which  are  in 
the  same  straight  line.  Find  how  many  lines  can  be  drawn, 
each  connecting  2  points. 

19.  In  how  many  ways  can  3  red  balls  and  2  white  balls  be 
selected  from  8  red  balls  and  5  white  balls  ? 

20.  In  an  algebra  class  of  25  students,  20  recite  each  day. 
In  how  many  ways  can  these  20  be  selected  for  one  day  ? 

21^' A  farmer  has  7  Borses.  In  how  many  ways  can  he 
hitch  a  two-horse  team  ? 

22.  In  an  examination  a  teacher  gives  a  pupil  the  choice  of 
any  8  questions  out  of  10.  In  how  many  ways  can  the  pupil 
choose  his  8  questions. 


9 


^ 


C. 


^  1° 


Ti 


n 


^/ 


CHAPTER  XXIII 
THE  BINOMIAL  THEOREM. 

238.  In  §  63  it  was  shown  that  any  positive  integral  powier, 
of  a  binomial  can  be  written  clown  by  some  laws  which  taken 
collectively  constitute  the  binomial  theorem. 

Thus,  by  these  laws, 

{a'  +  2by={ay  +  4{a'f{2b)  +  Q{a'y{2bY  +  4{a'){2by  +  (26)* 
=a^  +  8a^b  +  24a*b'  +  32aW  +  H)b\ 

In  the  general  case,  it  will  be  found  that  the  laws  in  §  63 
will  give 

It  will  be  recalled  that  no  rigorous  proof  of  this  theorem 
has  been  given.  The  binomial  theorem  can  be  proved  to 
hold  true  for  all  exponents,  integral  or  fractional,  positive  or 
negative.  We  shall  prove  the  theorem  here  for  the  case  when 
the  exponent  is  a  positive  integer. 

239.  Proof  when  the  exponent  is  a  positive  integer. 

From  the  rule  for  obtaining  the  product  of  two  expressions, 
which  is  based  upon  the  distributive  law,  it  follows  that  a  term 
in  the  product  of  any  finite  number  of  expressions  can  be  ob- 
tained by  multiplying  a  term  of  any  one  of  the  expressions  by  a 
term  from  each  of  the  other  expressions. 

A  repetition  of  this  process  in  every  possible  way  will  give  all 
of  the  terms  in  the  product  of  the  expressions. 

346 


THE  BINOMIAL  THEOREM  347 

For  example,  consider  the  product  {a  +  b){p  +  q){x  +  y).  In 
finding  the  product  ot  a  +  b  andp  +  g  we  multiply  each  term  of 
a  +  bhy  each  term  ot  p  +  q.  Each  of  these  resulting  terms  is  then 
multiplied  by  x  and  by  y  to  obtain  all  of  the  terms  in  the  product 
of  the  three  binomials.  This  process  evidently  amounts  to  the 
use  of  the  above  rule. 

.  Consider  the  expression  (a  +  Z>)^     This  may  be  written  in 
the  form 

{a  +  b)(a-\-b)(a-\-b) to  ^z  factors. 

If  we  select  a  tenn  from  each  of  the  n  factors  in  this  product^ 
and  multiply  these  terms  together^  and  do  this  in  every  pos- 
sible way^  we  shall  obtain  all  of  the  terms  in  the  product. 

(1)  Now  the  term  a  can  be  selected  from  all  of  the  factors 
in  just  one  way.  Hence,  the  product  of  these  a's,  which  is 
a'\  is  iho,  first  term  of  the  product. 

(2)  The  term  b  can  be  selected  from  one  factor  and  the  term 
a  from  each  of  the  other  n  —  1  factors ;  and  this  can  be  done  in 
as  many  ways  as  b  can  be  selected  from  the  n  factors,  which 
is  n  or  „  C^.  Hence,  a"~^6  can  be  selected  in  n  ways,  or  ^  C^ 
ways ;  that  is,  na'^~^b^  or  „  C^a'^~'^b^  is  the  second  ter^tn  of  the 
product. 

(3)  The  term  b  can  be  selected  from  each  of  two  factors  and 
the  term  a  from  each  of  the  other  n  — 2  factors  ;  and  this  can 
be  done  in  as  many  ways  as  two  5's  can  be  selected  from  the  n 
factors,  which  is  „  C^.  Hence,  a'^-'^b^  can  be  selected  in  „  C^  ways ; 
that  is,  „  C^a'^'^b'^  is  the  third  term  of  the  product. 

(4)  The  term  b  can  be  selected  from  each  of  three  factors 
and  the  term  a  from  each  of  the  other  n—^  factors;  and  this 
can  be  done  in  as  many  ways  as  three  ^'s  can  be  selected  from 
the  n  factors,  which  is  „  C^.  Hence,  a'^'W  can  be  selected  in 
„  C3  ways ;  that  is,  „  O^a/^'W  is  th-e  fourth  term  of  the  product. 


348  ALGEBRA 

(5)  Let  this  process  be  continued.  In  general,  the  term  h 
can  be  selected  from  each  of  r  factors  and  the  term  a  from 
each,  of  the  other  n—r  factors ;  and  this  can  be  done  in  as 
many  ways  as  r  i's  can  be  selected  from  the  n  factors,  which 
is  „(7^.  Hence,  a'^-^'W  can  be  selected  in  ^G^  ways;  that  is, 
^C/i^'-'lf  is  the  (r+l)^A  term,  of  the  product. 

(6)  Finally,  the  term  h  can  be  selected  from  all  of  the  fac- 
tors in  just  one  way.  Hence,  If  can  be  selected  in  just  one 
way  ;  that  is,  5"  is  the  last  term  of  the  product. 

Hence,  (aH-6)"=fl'*  +  „(?ia'^-^6-h„(?2a"-='6'+„(?3a""'6'+ 

+Xa"~'*6"+ +6"-  (1) 

The  expaiision  in  (1)  expresses  in  symbols  the  binomial 
theorem.  If,  in  this  identity,  „  Ci,  „  C^^  „  G^  etc.,  are  replaced  by 
their  values,  the  identity  becomes 

^  .  .  ^^n{n~1){n-2)-^  ■  ■  ■  („-.+  /)^_^^^  _^^„   ^^^ 

It  is  seen  that  (2)  conforms  to  the  laws  of  §  63. 

When  n  is  a  positive  integer  it  is  easily  shown  that  there  are 
always  /?  + 1  terms  in  the  expansion. 

When  r^  71,  the  coefficient  of  the  (r+l)th  term  (t.  6.,  the 
(n  +  l)th  term)  is  1 ;  but  when  r=?i  +  l,  the  coefficient  of  the 
(r  +  l)th  term  becomes 

n{n—l){n—'^) {n—n  — 1  +  1) 


n  +  1 


which  is  0,  since  the  last  factor  in  the  numerator  is  0. 
Hence,  a  term  does  not  exist  in  which  r  is  greater  than  n  ; 
that  is,  there  are  only  n-\-l  terms. 

Example  1.     Expand  (a^  +  i/^)^ 

Here  n=5.     And  fi^=^,  5^=10,  503=10,  ^C,=^. 


THE  EilNOMIAL  THEOREM  349 

Hence,       {x'  +  y'f={x'r  +  Mx^W)  +  ,C,{aff{yy  4-  .C^ix^Yifr  + 
\c\{x^){yy+(yr 
=ic"  +  5x^y^  +  lOx^y*  +  lOxV  +  5x^y^+  2/^^ 
Example  2.     Expand  (2— ar*/. 
Here,  ti=4.     And  ^0^=4,  ^0^=6,  fi^^^. 
Therefore,  {2-x^)'=i2y  +  ^{2f{-x^)+^C^{2fi-0(^Y  + 

,C3(2)(-^/  +  (-^r 
=lQ-32x^  +  2Ax^  -  %x^  +  x^\ 
Examples.     Expand  (a  — 2&  +  c)^ 
Grouping  terms,  this  becomes  [(a— 26)  +  c]^ 
Hence,  [{a-2h)  +  cY={a-2hf  +  fi^{a-2hfc-\-fi^{a-2hy  +  (^ 

=(a-26)»  +  3(a-26)2c  +  3(a-26)cHc^ 
Expanding  each  term,  we  have 

[(a-26)  +  c]^=a3-6a2&  +  12a&2-86^  +  3a='c-12a6c  +  126='c  +  3ac^- 

66cHc^ 

240.  The   general   term.     The   (r+l)th,   or  general  term, 

in(l)of  §239is„(?^a"-'-6'-,or-^ ^- ^— — ^ ' — 

By  substituting  the  values  of  a,  ^,  ?i,  r,  in  this  expression, 
for  the  (/•+l)th  term,  we  may  write  down  at  once  any  desired 
term  of  the  expansion  of  any  power  of  any  binomial. 

Example  1.    Find  the  8th  term  in  the  expansion  of  {2x—iy^. 

Here,  a=2x,  6=  — 1,  w=10,  r=7. 

Hence,  the  8th  term  =^^C,{2x)\-iy=^^C.,{2xf{-iy 

=i|^-8x»(-l)  =  -960a^. 

EXEBCISE  97. 

Expand : 

1.  {x-Vyy.  3.  (l  +  2aj2)*.  6.  (l-3a=')». 

2.  (2ic-3y)«.  4.  {^a'-Vhy.  6.  {x'-ay.. 


350  ALGEBRA 

7.  (i+2xy.  ^^  ^^_^i^e.        .  14.  (^^-wy. 

8.  (l-ixy,  ^^    (a-2+5-2)^  15.  ra-i  +  dy, 

9.  {x-^+xy.  /,j    2by  ^/  5 

10.  (2a;-2  +  l)^  -^^V  V^^"^/  *  16.'  {x'-y'^)\ 

17.  Find  the  third  term  in  the  expansion  of  (l  +  2a^y. 

18.  Find  the  sixth  term  in  the  expansion  of  (cc^  — 2y)^*'. 

_i        2 

19.  Find  the  eighth  term  in  tlie  expansion  of  (x  '^—a^y. 

20.  Find  the  fifth  term  in  the  expansion  of   (-^ — *^ 

21.  Find  the  fourth  term  in  the  expansion  of  (|«^^  +  |c~2)i^ 

-2.  2. 

22.  Find  the  seventh  term  in  the  expansion  of  {x   '^+x'^y^. 

f       V^  ^*^ 

23.  Find  tlie  sixteenth  term  in  the  expansion  of   {  1 


X 

24.  Find  the  twelftli  term  in  the  expansion  of  (2— la?^)'*. 

By  grouping  terms,  express  as  binomials  and  expand  : 
26.  {\-^x-xy,  27.  {2-a-\-hy.  29.  {ci~h-\-c-dy. 

26.  (£6^  +  2/'  +  ^')'.         28.  {X^x^-x^-xy.    30.  {2a-h^Zey. 

24 J.  Binomial  theorem,— exponent  negative  or  fractional. 
When  the  exponent  -is  a  negative  number  or  a  fraction  the  ex- 
pansion of  a  power  of  a  binomial  as  in  §  239  gives  an  inde- 
finitely large  number  of  terms.  This  follows  from  the  fact 
that,  for  such  an  exponent,  the  coeflBcient, 

n{n-V){n-2)  •  •  '\n-r^-\) 

of  the  general  term  can  never  become  zero. 

It  can  be  shown,  however,  that  the  use  of  the  binomial  theo- 
rem in  such  cases  is  allowable,  pravided  that  the  absolute 
value  of  a  is  greater  than  that  of  h.  It  is  advisable  to  exclude 
the  proofs  from  this  book.  The  student  may  assume  that  the 
theorem  holds  in  the  exercises  that  follow. 


THE  BINOMIAL  THEOREM  351 

Example  1.     Expand  {2—x^)-^  to  5  terms. 
We  have  i2-x')-^={2)-'+i-3)(2)-'{-x')  + 

(_3)(--4)r-5)(-6)^^^_,^_^y^ 

—  1    I      3   '«»2    1      3  /y.4    I      5  ^6    r      1  5  /y.8    I 

—  ¥   '   T'S^*^   ^TS"^    ^H'S'^   ^TS'^'^    ^ 

_3 

Example  2.-  Expand  (a  +  2a?)*  to  5  terms.  ^ 

We  have  {a  +  2x)^={a)^ -\-{f){a)~^{2x)+^i^i:^{a)~^{2xy  + 

ci)(z 

=a*+|a  ^a?— fa  ^a^^  +  ^a  ^o?-*^— jVgCt    ^  a?*+ 

1  2-  O  ,    ^ 

Example  3.     Expand  (i—x)  ^  to  4  terms. 

We  have  (l.-a?r^==(l)"^  +  (-i)(ir"2(-a^)  +  ti)H)(l)-|(_ic) 
4-tMJ.Kd)(l)-i(-;rf 


•    (iK=i^t:4)(«)-l-(2^)3^(J0G_^  .  .  .  . 


'  + 


=l+4a?+|a?'.+  A^  + 


EXERCISE  98. 

Expand  to  four  terms : 

1     /I  — oA-2  2  4 

^       ^     *  6.  (1-x'y.  10.  (ic  +  2r's-. 

3.  K-.r^  ''  ^^^'^^^'  .  11.  (8  +  .)i 

4.  (2x.-3y)-^.  8-  (20.^-1)"^. 

6.  (1  +  a;^)-^  9.  (i  +  a;)"^  12.  (2-ic2)-J. 

In  its  expansion  find  : 

13.  Tlie  6th  term  of  (2-a;)-3. 


352  ALGEBRA 

14.  The  10th  term  of  (a'^c^)^. 

15.  The  5th  term  of  {ei-e'^)-^. 

16.  The  8th  term  of  {l-2x^yi. 

242 .  Extraction  of  roots  by  use  of  the  binomial  theorem.  The 
binomial  theorem  may  be  used  to  extract  roots  of  arithmetical 
numbers.     The  process  is  best  shown  by  an  example. 

Example  1.    Find  to  4  places  of  decimals  the  value  of  y'W. 


(i)H)H)(25)-V-(_3).+    .  . 


2-  3         9  81 


80     6400     1024000 

=2-.0B75-.0014-.0001- 

=1.9610  approximately. 

A  similar  process  may  be  used  in  any  case.  Hence  the  fol- 
lowing rule : 

^Separate  the  number  into  two  parts^  the  first  of  ichich  is  the 
nearest  possible  perfect  power  of  which  the  required  root  can  be 
found ;  then  expand  the  resulting  binomial  by  the  binomial 
theorem^  and  combine  the  values  of  the  terms  thus  obtained. 

Thus,  v65  =  v'64  +  l  =  (43  +  l)^;   i>623=|/625-2=(54-2)*. 

It  is  evident  that  the  first  few  l^erms  of  the  expansion  will 
give  a  close  approximation  to  the  value  of  the  root,  if  the  suc- 
cessive terms  decrease  in  value  rapidly  ;  i.e.^  if  the  second 
term  of  the  binomial  is  much  smaller  than  the  first.  By 
§  241,  no  correct  approximation  of  the  root  can  be  found  by 


.  THE  BINOMIAL  THEOREM  353 

this  method  unless  the  given  number  is  expressed  as  a  bino- 
mial in  which  the  second  term  is  less  than  the  first. 

Note. — A  shorter  method  of  finding  the  approximate  value  of  any 
root  of  a  number  is  by  the  use  of  logarithms,  discussed  in  Chap- 
ter XXVI. 

EXERCISE  99. 

Find  to  four  places  of  decimals  the  value  of : 

1.  1^15.         3.  v2T9.         5.   vMT.         7.  1/3120.        9.  t>730. 

2.  1/240.       4.  yd.  6.  y^M.  8.  i/26.  10.  1/21. 

23 


CHAPTER  XXiy. 
PROGRESSIONS. 

243.  Series.  A  succession  of  terms,  in  which  each  term  after 
the  first  may  be  obtained  from  one  or  more  of  the  preceding 
terms  by  some  fixed  law ;  ^.e.,  obtained  in  the  same  way  for  all 
terms,  is  called  a  series. 

Thus,    2  +  4  +  6  +  8  +  10  +  12+ is  a  series.     Each    term 

after  the  first  may  be  obtained  by  adding  2  to  the  preceding  term. 

Also,  l  +  x-]-x^  +  oc^  +  x*+  '  ■  '  •  is  a  series.  Each  term  after 
the  first  may  be  obtained  by  multiplying  the  preceding  term  by  x. 

A  finite  series  is  one  that  has  a  finite  number  of  terms. 

An  infinite  series  is  one  that  has  an  infinite  number  of  terms. 

A  series  is  called  convergent  either  when  the  sum  of  all  of 
the  terms  equals  a  fixed  finite  number ;  or  when  the  sum  of 
the  first  n  terms  approaches  a  certain  fixed  number  as  a  limit, 
when  n  is  indefinitely  increased. 

A  series  is  called  divergent  when  the  sum  of  the  first  n 
terms  can  be  made  greater  than  any  assigned  number  which 
may  be  as  great  as  we  please,  by  taking  n  sufficiently  great. 

A  finite  series  is  ahoays  convergent. 

In  the  series  2  — 2  +  2  — 2  +  -  •  •  •,  the  sum  of  the  first  n 
terms  is  either  2  or  0  according  as  7i  is  odd  or  even.  Such  a 
series  is  called  an  oscillating  series. 

244.  We  shall  discuss  in  this  chapter  three  special  forms  of 
the  simpler  series,  known  as  arithmetical  progressions,  geometri- 
cal progressions,  and  harmonical  progressions. 

354 


PROGRESSIONS  355 

ARITHMETICAL  PROGRESSIONS. 

245.  An  arithmetical  progression  is  a  series  of  terms  in 
which  the  difference  between  any  term  and  the  preceding 
term  is  the  same  for  all  terms  of  the  series.  This  difference  is 
called  the  common  difference,  and  may  be  either  a  positive  or  a 
negative  number,  integral  or  fractional.  The  name  arithmeti- 
cal progression  is  usually  abbreviated  to  A.  P. 

Thus,  the  series  1  +  3  +  5  +  7  +  9  +  11+  •  •  •  •  •  is  an  A.  P.  in 
which  the  common  difference  is  2. 

And  the  series  12  +  8  +  4  +  0-1-8-12—14—  •  •  •  is  an  A.  P. 
in  which  the  common  difference  is  —4. 

246.  The  nth  term  of  an  A.  P. 

Let  a  stand  for  the  first  term  of _an  A.  p., 

/  for  the  nth  term, 
and  d  for  the  common  difference. 

Then,  by  the  definition  of  an  A.P., 

the  second  term  =  a-\-d, 
the  third  term  =  a-\r'2id^ 
the  fourth  term  =  a  +  3c?, 
the  fifth  term  —  a  +  4id^  etc. 

It  is  evident  from  these  expressions  that  the  coefficient  of  d 
in  the  expression  for  any  term  is  less  by  one  than  the  number 
of  the  term. 

Hence,  the  nth  term  =  a-{'(n  —  l)d; 

that  is,  /=a  +  (n—l)(f.  Formula  A. 

This  equation,  or  formula,  will  enable  us  to  find  the  value  of 
any  one  of  the  four  numbers,  I,  a,  n,  d,  if  the  values  of  the 
other  three  are  known. 


356  ALGEBRA 

Example  1.     Find  the  20th  term  of  the  series  4  +  7  + 10  + 13  +  •  •  • . 
Herea=4,  d=3,  n=20. 
Substituting  in  formula  A,  we  have 

Z=4  +  (20-l)3=61. 

Example  2.     The  4th  and  15th  terms  of  an  A.  P.,  are  9  and  31, 
respectively.     What  is  the  series  ? 

Here  the  4th  term  is  a  +  3d=9,  (1) 

and  the  15th  term  is    a  +  14<i=31.  (2) 

Solving  the  system  (1),  (2),  we  get  a=3,  and  d=2. 

Hence,  the  series  is 

3  +  5  +  7  +  9  +  11  +  13+ 

Examples.     In  the  series  5  +  9  +  13  +  17+ ,  what  term 

is  65  ? 

Here  a=5,  cZ=4,  Z=65,  n  is  unknown. 
Substituting  in  formula  A,  we  have 

65  =  5  +  (n-l)4. 
Solving  this  for  n,  we  get  11=16. 
Hence,  65  is  the  16th  term. 

247.  Sum  of  n  terms  of  an  A.  P.     Let  the  sum  of  n  terms  of 
an  A.  P.  be  represented  by  jS. 

Then  ^=a  +  {a  +  d)  +  (a  +  2d)+(a-\-Sd)+   •  •  •   +(l-d)+L 
Written  in  the  reverse  order,  we  have 

^=l+(l-d)  +  {l-2d)-i-(l-dd)+  •  •  •  •  +(a  +  d)  +  a. 

Adding  the  corresponding  terms  of  these  two  series,  the  d''s 
are  destroyed,  and  we  have 

2S=(a  +  l)'i-(a  +  l)  +  (a  +  ^+  •  •  •  to  7i  terms 
=  n(a-\-l). 
Hence,  *    S==^n(a  +  /).  Formula  i?. 

But,  by  formula  A^  l=a  +  (n—l)d. 


PROGRESSIONS  357 

Substituting  this  value  of  I  in  formula  B^  we  get 

5=^/7{2a  +  (/7-l)^}.  Formula  C. 

The  five  numbers,  a,  d^  /,  /j,  aS,  of  an  A.  P.,  we  shall  call  the 
elements  of  the  series. 

If  any  three  of  the  five  elements  of  an  A.  P.  are  knoAvn,  the 
values  of  the  other  two  may  be  found  by  use  of  formulas  A^ 
B,  and  G. 

Example  1.     Find  the  sum  of  25  terms  of  the  series 
-9-5-1  +  3  +  7+    •   •   •   •  . 
Here,  a=— 9,  <i=4,  n=25. 
Substituting  in  formula  C  gives 


>S=i-25-|  2(-9)  +  (25-l)4  1=975. 


Example  2.     The  first  term  of  an  A.  P.  is  2,  and  the  sum  of  20 
terms  is  135.     Find  the  common  difference. 
Here,  a=2,  n=20,  iS=135,  d  is  unknown. 
Froin  formula  C  we  have 

135=10(4  +  19d)., 
Solving  this  for  d,  we  get  d=^. 

Example  3.    How  many  terms  of  the  series  3  +  1  —  1  —  3—5—  •  •  • 
must  be  taken  to  make  —140? 

Here,  a=3,  d=—2,  S=  —  l'iO,  n  is  unknown. 
From  formula  C  we  have 

-U0=^n{Q  +  {7i-l)(-2)}. 

Simplifying  this  gives  the  equation 

n^-4n-U0=0. 

The  solutions  are  14  and  —10.     The  negative  solution  has  no 
meaning  in  this  problem,  hence  14  terms  must  be  taken. 


358  ALGEBRA 

Example  4.     In  a  certain  A.  P.  the  common  difference  is  1^, 
and  the  12th  term  is  12^.     Find  the  sum  of  the  first  7  terms. 
We  first  find  a  from  formula  A,  then  find  S  from  formula  C. 
From  formula  A, 

12^=a  +  lll|. 
Hence,  a=— 4. 

Then  from  formula  O, 

Example  5.     Find  the  sum  of  all  the  numbers  between  100  and 
500  which  are  multiples  of  6. 
The  numbers  between  100  and  500  which  are  multiples  of  6  are 
617,  6-18,  619,   •  •  •   •  6-83. 

Hence,  the  sum  is 

6-17  +  618  +  6194- +6-83, 

or  6(17+18  +  19+    •   •   •   •   •   +83). 

The  sum  of  the  A.  P.  enclosed  in  parentheses  is  obtained  by 
formulas  A  and  C. 

From  formula  A,  83=17+ (n-l)l, 

whence  7i=67. 

From  formula  O,  >9=-y  (34  +  66)=3350. 

Hence,  the  required  sum  is  6-3350,  or  20100. 

Example  6.     Show  that  the  sum  of  r  + 2*-^ +  3'+ n^ 

=inin  +  l)i2n  +  l). 

Note. — While  this  series  is  not  an  A.  P.  it  illustrates  an  application 
of  it. 

Since  (x-\-lf—ocr^=3x^  +  Sx  +  l  is  true  for  all  values  of  .t,  we  may- 
give  to  £c  a  succession  of  values  1,  2,  3,  etc.,  and  by  adding  the  n 
identities,  obtain  the  given  series.     That  is,    from 

(ic  +  1)*— a?^=3a?H3a?+l,  we  have 
when  x=l,  2'-l=»=3r  +  31  +  l, 

when  x=2,  S^-2^=3-2^  +  S2  +  l, 

when  a?=3,  4^— 3^=3-32  +  3-3  +  l, 


when  x=n,  (ii  +  l)^— n''=3-nH3-n  +  l. 


PROGRESSIONS  359 

Now  adding  columns,  and  observing  that  the  second  term 
of  each  left  hand  member  cancels  the  first  term  of  the  member 
above  it,  we  have 

(?i+l)^-P=3(P  +  22  +  3H   •  •   •  •  w'')  +  3(l  +  2  +  3+    •  •  •  •  n)  + 
(1  +  1  +  1+    •   •   •  •  to  n  terms) 
=3(P  +  2'^  +  3^+    •   •   •   •  n')+pi(n+l)  +  n.  _ 
Solving  for   12  +  2^  +  32+    •   •   •   •  »^^    we  obtain  |n(n  +  l)(2n  +  l). 
(Let  the  pupil  show  how  this  solution  was  obtained.) 

248.  Arithmetical  means.  If  thr^e  numbers  form  an  A.  P., 
the  middle  term  is  called  the  arithmetical  mean  of  the  other 
two  terms. 

Thus,  in  the  series  3  +  8  +  13,  8  is  the  arithmetical  mean  of  3 
and  13. 

If  ic  is  the  arithmetical  mean  of  a  and  b,  then,  by  the  defini- 
tion of  an  arithmetical  progression, 


a  +  b 
whence,  x=    ^    - 

Hence,  the  arithmeiical  tnean  betioeen  two  numbers  equals  half 
their  sum. 

249.  In  an  A.  P.  all  the  terms  between  any  two  terms  are 
called  the  arithmetical  means  of  those  two  terms. 

Thus,  since  2  +  4  +  6  +  8  +  10  is  an  A.  P.,  4,  6,  and  8  are  all 
arithmetical  means  of  2  and  10. 

Any  number  of  arithmetical  means  may  be  inserted  between 
any  two  given  numbers. 

Example  1.     Insert  7  arithmetical  means  between  10  and  30. 
Since  there  are  to  be  7  arithmetical  means,  30  must  be  the  9th 
term  of  an  A.  P.  of  which  10  is  the  first  term. 
Hence,  we  have  a=10,  /=30,  n=9. 


360  ALGEBRA 

Substituting  in  formula  A,  we  have 

30=10 +  8d. 
Solving,  d=2l. 

Hence,  the  required  series  is 


EXERCISE  100. 

1.  Find  the  30th  term -in  1  +  6  +  11  +  16+ 

2.  Find  the  16th  term  in  -8-5—2  +  1  +  4+ 

3.  Find  the  23rd  term  in  — |— 1-  +  ^  +  |+ 

4.  Find  the  54th  term  in  11  +  17  +  23+   •••  •. 

5.  The  6th  term  of  an  A.  P.  is  17,  and  the  15tli  term  is  44 
Pind  the  common  difference. 

6.  The  3rd  term  of  an  A.  P.  is  0,  and  the  9th  term  is  22. 
Find  the  common  difference. 

7.  The  fifth  term  of  an  A.  P.  is  21,  and  the  8th  term  is  33. 
What  is  the  12th  term  ?     The  20th  ? 

8.  The  2nd  term  of  an  A.P.  is  7,  and  the  11th  term  is  20^. 
What  is  the  7th  term?     The  15th  term? 

9.  Which  term  of  the  series  1  +  4  +  7  +  10+  •  •  •  is  46  ? 

10.  Which    term   of    the    series   10  +  6^+3-1- 

is  -25? 

11.  In  an  A.  P.  whose  common  difference  is  6,  the  11th  term 
is  72.     What  is  the  first  term  ? 

12.  Find  the  sum  of  the  first  twenty  terms  of  the  series 
42  +  39  +  36 

13.  Find  the  sum  of  the  first  thirteen  terms  of  the  series 
8  +  12  +  16+ 

14.  Find  the  sum  of  the  first  fifty  odd  numbers. 


PROGRESSIONS  361 

15.  Find  the  sum  of  the  first  fifty  even  numbers. 

16.  Find  the  sum  of  the  first  ten  terms  of  a  series  whose 
first  term  is  —6  and  tenth  term  25i. 

17.  How  many  terms  of  the  series  15  +  12+9+  •  •  •  •  niust 
be  taken  to  make  45  ? 

18.  In  an  A.  P.  whose  first  term  is  8,  the  sum  of  tlie  first  9 
terms  is  324.     What  is  the  9th  term  ? 

19.  Find  the  sum  of  all  odd  numbers  of  two  digits. 

20.  Find  the  sum  of  all  even  numbers  between  100  and  300. 

21.  Find  the  sum  of  all  numbers  between  50  and  250  which 
are  divisible  by  4. 

22.  Show  that  the  sum  of  the  first  n  odd  numbers  is  71"^. 

23.  Show  that  r+2^  +  3^+ n'=  ||(^  +  1)  V 

Suggestion.  See  Example  6,  §247.  Remember  that  (ic+1)*— £c*= 
4a73+6a?2+4£t?+l. 

24.  Insert  four  arithmetical  means  betweeji  5  and  25. 

25.  Insert  six  arithmetical  means  between  —10  and  \. 

26.  Find  three  numbers  which  are  in  A. P.,  such  that  their 
sum  is  18,  and  such  that  the  product  of  the  first  and  last  is 
greater  than  the  second  by  14. 

27.  A  man  had  a  cistern  dug  12  feet  deep.  The  first  foot 
cost  $1,  the  second  11.25,  the  third  $1.50,  and  so  on.  What 
did  the  digging  cost  ? 

28.  A  body  falls  toward  the  earth  at  the  rate  of  a  feet  the 
first  second,  3a  feet  the  second  second,  5a  feet  the  third  sec- 
ond, and  so  on.     How  far  will  it  fall  in  t  seconds  ? 

29.  A  man  pays  $50  of  a  debt  the  first  year,  $75  the  second 
year,  $100  the  third  year,  and  so  on.  In  this  way  he  pays  the 
whole  debt  of  $1100.     How  many  years  does  it  require  ? 


362  ALGEBRA 

GEOMETRICAL  PROGRESSIONS. 

250.  A  geometrical  progression  is  a  series  of  terms  in  Avhicli 
the  ratio  of  any  term  to  the  preceding  term  is  the  same  for  all 
terms  of  the  series.  This  ratio  is  called  the  common  ratio, 
and  may  be  either  positive  or  negative.  The  name  geometrical 
progression  is  usually  abbreviated  to  G.  P. 

Thus,  the  series  1  +  2  +  4  +  8  +  16+  •  •  •  ■  is  a  G.  P.  in  which 
the  common  ratio  is  2. 

The  series  2— 1  +  1  —  1  +  1—3^4-  •  •  •  •  is  a  G.  P.  in  which  the 
common  ratio  is  —  |. 

In  either  series,  if  any  term  be  multiplied  by  the  common  ratio, 
the  product  will  be  the  next  term. 

251.  The  nth  term  of  a  G.  P. 

Let  a  stand  for  the  first  term  of  a  G.  P. ; 

I  for  the  nth  term ; 
and  r  for  the  common  ratio. 
Then,  by  definition  of  a  G.  P., 

the  second  term  =  ar, 
•  the^  third  term  —  ar'^^ 

the  fourth  term  =  ar^^QiG. 
It  is  evident  from  these  expressions  that  the  coefficient  of  a 
in  any  term  is  r,  with  an  exponent  less  by  one  than  the  num- 
ber of  the  term. 

Hence,  the  nth  t€rm  =  ar''-^  \ 

that  is,  l=:ar"-^.  Formula  yl. 

This  equation,  or  formula,  will  enable  us  to  find  the  value 
of  any  one  of  the  four  numbers,  Z,  a,  r,  n,  if  the  values  of  the 
other  three  are  known. 

Note. — Since  n  is  an  exponent,  its  value  cannot  in  general  be  found, 
except  by  inspection,  without  the  use  of  logarithms.  See  Chapter 
XXVI.  In  all  of  the  problems  in  this  chapter  n  can  be  found  by 
inspection. 


PROGRESSIONS  363 

Example  1 .     Find  the  nioth  term  of  the  series  2  +  6  +  18+ 

Here  a=2,  r=3,  n=9. 

Hence,  Z=2-3«=13,122. 

Example  2.     The  tenth  term  of  a  G.  P.  is  ^-f^,  and  the  first 
term  is  1.     Find  the  common  ratio. 

Here  a=l,  n=10,  1=^\y- 

Hence,  -^-g  =lr^j 

and  r=}. 

Example  3.     The  fifth  and  eighth  terms  of  a  G.  P.  are  ^f  and 
— Ill,  respectively.     Write  the  series. 

Here  the  fifth  term  =ar*=^,  (1) 

and  the  eighth  term    =ar^=— |ff.  (2) 

Dividing  (2)  by  (1),  r'=-^; 

whence  *'=— I- 

Replacing  r  by  -|  in  (1),      aff =ff  ; 
whence  a =3. 

Therefore,  the  series  is 

3        0i    4 8    I    1  6 33    I      64    128    i 


^'^5^^fc 


252.  Sum  of  n  terms  of  a  G-.  P.     Let  the  sum  of  n  terms  of 
a  G.  P.  be  represented  by  /S. 

Then,  jS=a  +  ar'\-ar''  +  ar'-{- +ar"-2  +  ar»-\ 

=  a(l-^r  +  r'  +  r'+   •  •  •   +r"-2+-^n-i^ 

=a(l:Z^\  §76. 


1-r  ^ 

Hence,  5=-^^—^.  Formula^. 

1— r 

But,  by  formula  ^,  l=ar''-^. 

Hence,  from  formula  A  and  formula  B,  we  get 

A     a—r/ 


1-r' 

The  five  numbers,  a,  r,  ?i,  /,  /^,  of  a  G.  P.  we  call  the  elements 


364  ALGEBRA 

If  any  three  of  the  five  elements  of  a  G.  P.  be  known,  then 
the  other  two  may  be  found  by  the  use  of  formulas  A,  B^ 
and  C. 

Example  1.     Find  the  sum  of  the  first  six  terms  of  the  series 

.4M-64-9  +  -V-+ 

We  have  a=4,  r=|,  /i=6. 

Hence,  ^^4{l-(fr}^ 

Example  2.  The  first  term  of  a  G.  P.  is  3,  the  sixth  term  is 
9375,  and  the  sum  of  the  first  six  terms  is  11,718.  What  is  the 
common  ratio  ? 

By  formula  (7,  • 

•11,718  =  3=^^; 

1— r     ' 
whence  r=5. 

253.  Sum  of  an  infinite  number  of  terms  of  a  G.  P. 

From  §  252  we  have 

\—r 

This  may  be  written  in  the  form 


S=- 


1—r     \—r 

Now,  if  r  he  less  than  1,  r'*  will  be  less  still,  and  by  increas- 
ing n  sufficiently,  r"  can  be  made  less  than  any  assigned  value 
which  we  may  take  as  small  as  we  please. 

Thus,  if  r=i,  t^^tV,  ^''=^4,  ^'^fK,  '^'=-toW,  ^"=1™,  etc. 
Now,  at  the  same  time  that  r^  becomes  less  than  any  as- 

signed  value,  ^ will  also  become  less  than  any   assigned 

value,  however  small.     Hence,  if  n  be  taken  sufficiently  great, 

5 will  approach  indefinitely  near  in  value  to  = . 

1—r     1—r  1—r 


PROGRESSIONS 


'6^)0 


Consequently,  in  a  G.  P.  where  the  common  ratio  is  less  than 
1,  by  taking  71  sufficiently  great,  the  sum  of  n  terms,  can  be 

made  to  differ  from  :^—--  by  a  number  less  than  any  assigned 

value,  which  may  be  taken  as  small  as  we  please. 

Hence,  the  sum  of  an  infinite  number  of  terms  of  a  G.  P, 
ichose   common   ratio    is    less    than    1    is  defined  as    the   limit 

^*         That  is,  when  n  is  infinite  and  r  is  less  than  1, 


1-r 

S=A-'  Formula  J9. 

1 — r 

Example  1.     Find  the  sum  of  the  infinite  series  of  terms 
9  +  6  +  4+    •   •   •. 

Here  r=|;  hence,  the  formula  >S=T-—   may  be-  applied.     We 

9 
have  ^=3— — g=27. 

This  formula  can  be  used  to  find  the  value  of  a  repeating 
decimal. 

Example  2.     Find  the  value  of  .3333 


This  is  a  G.  P.  Avhose  first  term  is  y\,  and  common  ratio  ^V- 
Hence,      ^=-^=1=,]. 

Example  3.     Find  the  value  of  .12232323 

We  have. 12232323 =TVo+To¥oT7+ro¥o^+To^¥oiroo  +  - 


Hence,  .12232323- 


•Too  T^-J-9  00— 9  900' 


254.  Geometrical  means.  If  three  numbers  form  a  G.  P., 
the  middle  term  is  called  the  geometrical  mean  of  the  other 
two  terms. 


366  ALGEBRA 

Thus,  in  the  series  4  +  12  +  36,  12  is  the  geometrical  mean  of  4 
and  36. 

If  X  is  the  geometrical  mean  of  a  and  ^,  then  by  the  defini- 
tion of  a  geometrical  progression,  we  have 

h     X 


Solving,  x=i/ab. 

Hence,  the  geometrical  mean  of  two  numhers  equals  the  square 
root  of  their  product. 

255.  In  a  G.  P.  all  of  the  terms  between  any  two  terms  are 
called  the  geometrical  means  of  those  two  terms. 

Thus,  since  1  +  3  +  9  +  27  +  81  is  a  G.P.,  3,  9,  and  27,  are  geomet- 
rical means  of  1  and  81. 

Any  number  of  geometrical  means  may  be  inserted  between 
any  two  given  numbers. 

Example  1.     Insert  three  geometrical  means  between  ^  and  32. 
Since  there  are  to  be  three  geometrical  means,  |  must  be  the 
first  term,  and  32  the  fifth  term,  of  a  G.  P. 
Hence,  by  formula  A  we  have 

ir*=32; 
whence  r=4. 

Therefore,  the  required  series  is 

1  +  ^  +  2  +  8  +  32. 

EXERCISE    101. 

1.  Find  the  ninth  term  of  the  series  1  +  6  +  36+ 

2.  Findthetenth  term  of  the  series  Jg—i  + 1—4+ 

3.  Find  the  eighth  term  of  the  series  2  +  3  + 4i+ 


PROGRESSIONS  367 

4.  Find  the  twelfth  term  of  the  series  1 2  +  -^4~  *  '  *  '   • 

X         X 

5.  The  second  term  of  a  G.  P.  is  i,  and  tlie  eighth  term  is 
2^g.     Find  the  tenth  term. 

6.  Tlie  first  term  of  a  G.  P.  is  3,  and  the  tliird  term  is  6. 
Find  the  sixth  term. 

7.  The  second  term  of  a  G.  P.  is  3,  and  the  fifth  term  is  ^-j-. 
Find  the  fourth  term. 

8.  The   common  ratio  of  a  G.  P.  is  3,  and  the  seventh  term 
is  81.     Find  the  first  term. 

9.  The  third  term  of  a  G.  P.  is  i,  and  the  eighth  term  is  128. 
Write  the  first  eight  terms. 

10.  Insert  two  geometrical  means  between  125  and  —8. 

11.  Insert  three  geometrical  means  between  1  and  4. 

12.  Insert  five  geometrical  means  between  2  and  ||-. 

13.  Find  the  geometrical  mean  of  6  and  96. 

Find  the  sum  of : 

14.  Five  terms  of  i  +  i  +  |+ 

15.  Twelve  terms  of  2—4  +  8- 

16.  Ten  terms  of  If +  22+5/3 

17.  Six  terms  of  64-32  +  16-  ..••.. 

18.  Ten  terms  of  -f +i.-5_|_  ...... 

19.  In  a  G.  P.  whose  first  term  is  1,  and  common  ratio  4, 
how  many  terms  must  be  added  to  make  21  ? 

20.  In  a  G.  P.  whose  first  term  is  1,  and  common  ratio  —  i, 
how  many  terms  must  be  added  to  make  |^  ? 

Find  the  sum  of  an  infinite  number  of  terms  of : 
,21.  15  +  5  +  1+  ...... 

22.  -3-^i-^V+ • 


368  ALGEBRA 

23.  I.  +  1  +  -J+ , 

24.  i  +  i  +  i+ •  . 

25.  5-3  +  1- 

26.  In  a  G.  P.  the  common  ratio  is  i.  What  must  be  the 
first  term  in  order  that  the  sum  of  an  infinite  number  of  terms 
may  be  80  ? 

27.  In  a  G.  P.  the  first  term  is  5.  What  must  be  the  com- 
mon ratio  in  order  that  the  sum  of  an  infinite  number  of  terms 
may  be  4y<Y  ? 

28.  Find  the  G.  P.,  the  sum  of  an  infinite  number  of  terms 
of  which  is  2,  and  whose  second  term  is  |. 

29.  Find  the  G.  P.,  the  sum  of  an  infinite  number  of  terms 
of  which  is  27,  and  whose  second  term  is  —12. 

Find  the  values  of  : 

30.  .6666  •  •  •  .     32.  .150150  •  •  •  .    34.  1.45151  •  •  •  . 

31.  .2727  •  •  •  .     33.  .19999  •  •  •  .      35.  .16666  •  •  •  . 

36.  Sliow  that  if  all  of  the  terms  of  a  G.  P.  be  multiplied  by 
the  same  number,  the  resulting  terms  will  form  a  G.  P. 

37.  If  a  body  moves  |  ft.  the  first  second ;  f  ft.  the  second 
second ;  lift,  the  third  second ;  and  so  on ;  how  far  will  it 
travel  in  10  seconds  ? 

38.  A  man  saves  each  year  twice  as  much  as  the  preceding 
year,  and  he  saves  $150  the  first  year.  How  long  will  it  take 
for  him  to  save  $2250  ? 

39.  To  what  sum  will  II  amount  at  5%  compound  interest 
in  4  years  ? 

Suggestion.     a=$l,  r=1.05,  7i=5. 

40.  A  ball  falls  from  a  height  of  100  feet,  and  rebounds  after 
each  fall  one-fifth  of  the  distance  it  fell.  Through  what 
distance  will  it  have  traveled  at  the  end  of  the  fifth  fall  ? 


PROGRESSIONS  369 

HARMONICAL  PROGRESSIONS. 

256.  If  the  series  aH-^  +  c  +  <^+  •  •  •  •  is  an  arithmetical 
progression,  the  series    -  +  7,  +  -+  7+   *  *  •  •  is   called   a  har- 

monical  progression.  That  is,  a  series,  each  term  of  which  is 
the  reciprocal*  of  the  terms  that  form  an  arithmetical  progres- 
sion, is  a  harmonical  progression.  The  name  har  monical 
progression  is  usually  abbreviated  to  H.  P. 

Thus,  since  1  +  3  +  5  +  7+ is  an  A.  P.,  the  series  1  +  |  + 

KH- isaH.  P.      - 

Also,  ^+1 +  l  +  A  +  i  +  T9+  •  •  •  is  a  H.  P.,  because  2  +  |  +  5  + 
-V-  +  8+y+ is  an  A.  P. 

257.  Since  to  every  harmonical  progression  there  is  a  cor- 
responding arithmetical  progression,  problems  in  harmonical 
progression  may  generally  be  solved  by  inverting  the  terms, 
and  using  the  principles,  established  for  the  resulting  arith- 
metical progressions.        ' 

Thus,  any  tern)  of  a  II.  P.  may  be  found  by  obtaining  the 
same  term  of  the  corresponding  A.  P.,  and  inverting  it. 

No  formula  has  been'  developed  for  finding  the  sum  of  n 
terms  of  a  H.  P. 

Example  1.     Find  the  tenth  term  of  the  H.  P.  1  + 1  +  i  + 1  +  •  •  •  . 
Here  the  corresponding  A.  P.  is  1  +  1  +  2  +  1+    •   •   •   •  . 
By  the  method  of  §  246,  the  tentli  term  of  this  A.  P.  is   y . 
Inverting,  we  get  y\,  the  tenth  term  of  the  H.  P. 

258.  Harmonical  means.  If  three  numbers  form  a  H.  P., 
the  middle  term  is  called  the  harmonical  mean  of  the  other 
two  terms. 

Thus,  since  l  +  |  +  yV  is  a  H.  P.,  |  is  the  harmonical  mean  of  1 
and  yV- 

*  The  reciprocal  of  a  number  is  1  divided  by  that  number. 
24 


3Y0  ALGEBRA 


Let  X  represent  the  harmonical  mean  of  a  and   h,  then 
-  will  be  the  arithmetical  mean  of  -  and  -j- 


1  +  1 

Hence,  \a     b, 

X-    2     ' 

whence  ^=— i-r* 


259.  In  a  H.  P.  all  of  the  terms  between  any  two  given 
terms  are  called  harmonical  means  of  those  two  terms. 

Thus,  since  -Ki  +  i  +  TV  +  iV  is  a  H.  P.,  |,  ^  Vt.  are  harmonical 
means  of  |  and  yV- 

Any  number  of  harmonical  means  may  be  inserted  between 
any  two  given  numbers. 

Example  1.    Insert  four  harmonical  means  between  —  |  and  ^. 

We  must  first  insert  four  arithmetical  means  between  —  5  and 
10.  These  are  —2,  1,  4,  and  7.  Hence,  the  required  harmonical 
means  are  — ^,  1,  |,  and.|. 

260.  If  A,  G,  and  H,  represent,  respectively,  the  arithmeti- 
cal, geometrical,  and  harmonical  means  of  a  and  h,  then  by  the 
preceding  sections  we  have 


a  +  b 
Hence,  A-B:=—^^x—T-T=cib=G\ 

A  Oj-t  0 


Therefore, 


A  _G^ 

a~H 

That  is,  the  geometrical  mean  of  any  tioo  numbers  is  also 
the  geometrical  mean  of  their  arithmetical  and  harmonical 
means. 


PROGRESSIONS  371 

EXERCISE  102. 

1.  Find  the  seventh  term  of  the  series  1+i+TT^"  *  '  '  • 

2.  Find  the  twentieth  terra  in  the  series  — i— i— i+1 
+•  •  •   . 

3.  Find  the  fifteenth  term  in  the  series  — |— 2  +  24-|  +  -  •  •  •• 

4.  Find  the  twelfth  term  in  the  series  2  +  ^  +  y\  +  •  •  •  •  . 

5.  Find  the  H.  P.  in  which  the  third  term  is  ^  and  the 
seventh  term  is  ^. 

6.  Find  the  H.  P.  in  which  the  third  term  is  2  and  the 
thirty- second  term  is  ^j. 

7.  Find  the  harmonical  mean  of  i  and  yL . 

8.  Find  the  liarmonical  mean  of  —  f  and  |. 

9.  Insert  four  harmonical  means  between  —  f  and  |. 

10.  Insert  five  harmonical  means  between  i  and  3. 

11.  The  geometrical  mean  of  two  numbers  is  4,  and  their 
harmonical  mean  is  ^.     What  are  the  numbers  ? 

EXERCISES  FOR  REVIEW    (VII). 

1.  Define  permutation ;   combination. 

2.  What  symbol  represents  the  number  of  permutations 
of  n  things  taken  r  at  a  time  ?  What  represents  the  number  of 
combinations  of  n  things  taken  r  at  a  time  ? 

3.  What  is  the  value  of  ^P^  ?  Of  ,P,  ?  Of  ^^P^  ?  Of  „P^  ? 
Derive  the  formula  for  permutations. 

4.  In  how  many  ways  can  a  number  of  two  digits  be  made 
from  the  digits  1,  2,  3,  4  ? 


372  ALGEBRA 

5.  How  many  changes  can  be  rung  with  3  bells  out  of  6  ? 
How  many  with  the  whole  peal  ? 

6.  What  name  is  given  to  \n?    In  what  other  way  is   it 
sometimes  written  ?    What  does  it  mean  ?     Find  the  value  of 

|_3_;  |4 ;  |_6;  \n. 

7.  Find  the  number  of  permutations  of  the  letters  in  the 
the  words  United  States. 

8.  In  how  many  ways  can  the  letters  in  the  word  Cincin- 
nati be  arranged  ? 

9.  Find  the  value  of  76;;  20^3;  e^^;  nO^',  n^A  n^^w     Derive 
the  formula  for  combinations. 

10  Show  that,,  (7^=„(7„_,.     Find  .^Ogg. 

11.  In  how  many  Avays  can  1  boy  and  1  man  be  selected 
from  5  boys  and  5  men  ? 

12.  In  a  system  of  12  lines  lying  in  the  same  plane,  no  two 
of  which  are  parallel  and  no  three  of  which  pass  through  the 
same  point,  how  many  points  are  there  where  two  lines 
intersect? 

13.  Out  of  14  Democrats  and  20  Republicans,  how  many 
different  committees  can  be  formed  each  consisting  of  2 
Democrats  and  3  Republicans  ? 

14.  There  are  20  things  of  one  kind  and  10  of  another ;  how 
many  different  sets  can  be  made  containing  4  of  the  first  and  3 
of  the  second  ? 

1 5.  Show  by  use  of  combinations  that  {a + hf =a^-\-  Sa'^b  +  ^ab^ 
-Vb\ 

16.  What  is  the  binomial  theorem  ?    Illustrate. 

17.  How  did  we  prove  the  binomial  theorem  for  positive  in- 
tegral exponents  ? 


PROGRESSIONS  373 

18.  What  is  meant  by  the  expansion  of  a  power  of  a  bino- 
mial ?  If  the  exponent  of  the  binomial  is  fractional  or  nega- 
tive, how  many  terms  will  there  be  in  the  expansion  ? 

19.  Expand  {l-\x^y. 

20.  Expand  (1— a-)    »  to  four  terms. 

21.  What  is  the  expression  for  the  rth  term  of  {a  +  byi 

22.  Find  the  21st  term  of  (2-3£«;-\ 

23.  Find  the  6th  term  of  {\-^xy\ 

24.  Expand  {l^-x—x"^)^  by  the  use  of  the  binomial  theorem. 

25.  By  the  use  of  the  binomial  theorem  find  {a)  the  square 
root  of  8  ;  {p)  the  cube  root  of  25 ;  (c)  the  fifth  root  of  84. 

26.  What  is  an  arithmetical  progression  ?     Illustrate. 

27.  What  is  the  formula  for  the  7ith  term  of  an  A.  P.  ? 

28.  Find  the  20th  term  in  1  +  3  +  5  +  7+   •  •  •  •  . 

29.  Find  the  16th  term  of  the  series  l-i-2-  •  •  •  •  . 

30.  The  first  term  of  an  A.  P.  is  f  and  the  twentieth  term  is 
25.     Find  the  fifteenth  term  ;  the  fortieth  term. 

31.  The  seventh  term  of  an  A.  P.  is  5,  and  the  fifth  term  is  7. 
Find  the  first  term. 

32.  Write  the  formula  for  the  sum  of  n  terms  of  an  A.  P. 
How  is  it  derived  ? 

33.  Find   the  sum   of  fifteen  terms   of  the  series  2  +  5  +  8 

+••••. 

34.  The  second  term  of  an  A.  P.  is  21,  and  the  fifteenth  term 
is  22.     Find  the  sum  of  ten  terms. 

36.  What  is  meant  by  the  arithmetical  mean  between  two 


given  numbers.     Illustrate. 


374  ALGEBRA 

36.  Insert  3  arithmetical  means  between  4  and  20. 

37.  Write  the  formula  for  the  nth  term  of  a  G.  P. 

38.  What  is  the  42nd  term  of  the  series  2  +  3  +  41+  •  •  •  •  ? 

39.  The  third  term  of  a  G.  P.  is   8,  and  the  8th  term  is 
—  8192.     Find  the  first  term  and  the  common  ratio. 

40.  Write  the  formula  for  the  sum  of  n  terms  of  a  G.  P. 
How  is  it  derived  ? 


41.  Find  the  sum  of  12  terms  of  |-+i+| 


42.  Find  the  arithmetical  mean  of  2  and  21. 

43.  Insert  5  arithmetical  means  between  1  and  16. 

44.  Find  the  geometrical  mean  of  2  and  32. 

45.  The  arithmetical  mean  of  two  numbers  is  8,  and  their 
harmonical  mean  is  6.     Find  their  geometrical  mean. 

46.  If  the  arithmetical  mean  between  a  and  h  be  double  the 
geometrical  mean,  find  a^^b. 

47.  Insert  three  geometrical  means  between  2  and  162. 


CHAPTER   XXV. 

UNDETERMINED    COEFFICIENTS. 

261.  In  Chapters  XXIII  and  XXIV  we  have  had  four  ex- 
amples of  series,  each  developed  by  a  different  law. 

The  "binomial  series  is  developed  by  use  of  the  binomial 
theorem. 

The  arithmetical  progression  is  developed  by  adding  the  same 
quantity  to  each  term  to  get  the  following  term. 

The  geometrical  progression  is  developed  by  multiplying  each 
term  by  the  same  quantity  to  obtain  the  following  term. 

The  harmonical  progression  is  developed  by  developing  an 
arithmetical  progression,  and  inverting  its  terms. 

Series  may  originate  in  many  different  ways.  A  fraction 
may  sometimes  be  expanded  into  a  series  by  division. 

Thus, may  be  expanded  when  x  is  less  than  1.     Dividing 

X  "t~  3? 

1  by  1  +  07,  we  have 
1 


l—x  +  x^—x'^  +  x*— 

1  +  x 

A  surd  may  sometimes  be  expanded  into  a  series  by  extract- 
ing the  indicated  root. 


Thus,  y4c+x  may  be  expanded  in  ascending  powers  of  a?,  when 
X  is  less  than  4.  If  the  square  root  is  extracted  as  shown  in  the 
Appendix,  we  have 

■]/^-\-x=2  +  ix—^\x^  +  -^\^x^- 

One  of  the  most  important  elementary  methods  of  develop- 
ing a  series  is  based  upon  a  principle  known  as  the  Theorem  of 

875 


376  ALGEBRA. 

Undetermined  Coefficients.  This  principle  is  discussed  in  the 
following  sections. 

262.  Theorem  of  Undetermined  Coefficients. 

If  the   series   A-\-Bx+ Cx^ +  J)x^  +   •  •  •    'is   equal  to  the 
ssries  a  +  bx  +  cx'^-{-dx^+   '  •  •  •  ,  for   all   values    of  x   ichich 
make  both  series  convergent^  then  the  coefficients  of  like  powers 
ofx  in  the  two  series  must  he  equal ;  that  is^ 
A  =  a,  B=h,  C=c^  etc. 

We  give  the  proofs  of  this  principle  for  the  two  separate 
possible  cases. 

(1)  Both  series  finite.     Consider  the  equation 

ax''^hx''-^-Vcx''-^-\- +joa;+5'==0, 

in  which  the  series  is  written  in  descending  powers  of  x.  If 
tills  equation  is  an  identity,  it  is  satisfied  by  any  fi^iite  value 
that  may  be  given  x.  Let  a^,  a^,  ag,  •  •  •  •  a„,  be  n  different 
values  of  x  which  must  satisfy  the  equation.  Then,  the  first 
number  is  divisible  by  {x—a^.,  {x—a^.,  etc.  (§  95)  ;  therefore  the 
equation  may  be  written 

a{x—a^{x—a.^{x—a^ (ic— a„)  =  0. 

If,  now,  h  be  any  other  value  of  cc,  we  have 

a(b—a^(b—a^{b—a^ (^  — a„)=0. 

.  But,  since  h  is  different  from  a^,  a^j  ^^  etc.,  none  of  the 
binomial  factors  can  be  zero.  Hence,  since  the  product  of  all 
the  factors  is  zero,  a  must  equal  zero.  Now,  since  a=0, 
aa;"=0;  and,  rejecting  the  term  aic%  it  can  be  shown  in  like 
manner  that  J=0.     Rejecting  5ic"~\  we  get  c=0  ;  and  so  on. 

Hence,   ^/' aa;"  +  Jic"-^  +  cic"-^ px-\^q  =  0  is  to  he  true 

independently  of  the  value  ofx.,  i.  e.,for  all  finite  values  ofx^  then 
each  of  the  coefficients  must  he  zero. 


UNDETERMINED  COEFFICIENTS  377 

Now,   if  the   series   Ax*" -^  Bx""^ -\- Cx''-'^  + +P    is 

equal  to  the  series  aic'*  +  ^""^  +  ca;"~^  + +jt?,  for  all 

finite  values  of  x^  we  have 

Ax^'  +  Bx''-^^-  Cic"-2+  •  •  •  •  +P  =  aa;»  +  ^"-i  +  ca;"-2+ 4-jt? 

or     {A—a)x''^{B-h)x''-^+{C—c)x''-^-{' -\-{P—p)=0. 

And  since  this  equation  is  to  be  true  for  all  finite  values  of 
£c,  we  must  have 

^-a=0,  B-b=0,  C-c=0, j>-p=o, 

or  A=a,        B=b,        C=c,-      P=p. 

(2)  One,  or  both,  series  infinite.  In  the  convergent  infinite 
series  a  +  bx-\-cx-  +  dx^  +  ex*+fx^^gx^+  *  '  *  '?  which  is  written 
in  ascending,  positive,  integral  powers  of  x,  any  term  whose 
value  is  not  zero  may  be  made  greater  than  the  sum  of  all 
terms  that  follow  it  by  making  x  sufficiently  small. 

Let  us  choose  the  term  clx^.  Now  let  k  be  greater  than  any 
coefficient  following  d.  Then  kx\l  +  x  +  x^ -}-  x^  +  •  •  •  •  )  is 
greater  than  ex*  +  fx^  +  gx^ -{- Or,  since 

l  +  x  +  x'  +  x'^   •  •  •  •    =Y37-, 

X        X 

kx*YZ^  is  greater  than  ex*+fx^+gx^+ * 

1  kx 

But  dx^  is  greater  than  kx*YZZ~'>  i^  i  _    is  less  than  d. 

X       X         X       X 

Now,  since  d  is  not  zero, ..  _     can  be  made  less  than  d,  for 

by  taking  x  sufficiently  small,  the  numerator  may  be  made  less 
than  any  assigned  value,  while  the  denominator  will  approach  1. 

*  When  some  of  the  terms  are  negative  and  the  negative  signs  before 
the  terms  are  changed  to  positive  signs,  a  new  series  greater  than  the 
oUl  arises;  lience  if  the  principle  is  proved  for  all  terms  positive,  it  is 
evidently  true  when  part  of  the  terms  are  negative. 


378  ALGEBRA 

Hence,  dx"  can  be  made  greater  than  ea;*+/£c^  +  <7£«^-f 

In  like  manner,  ayiy  term  whose  value  is  not  zero  in  the 
above  series  can  be  made  greater  than  the  sum  of  all  the  fol- 
lowing terms. 

If,  now, 

A^Bx^Cx^^Dx"-^ =a-\-hx-^cx^-^dx^\ 

be  true  independently  of  the  value  of  a?,  we  have 

{A-a)^{B-h)x-^{C-c)x^-V{D-d)x'^-\-   •  •  •  •  •   =0. 

Now,  A— a  must  be  either  zero  or  not  zero;  and  if  ^— a 
were  not  zero,  by  taking  x  sufficiently  smalL^— «  could  be 
made  greater  than  the  sum  of  all  terms  that  follow.  But  this 
can  not  be  possible,  since  the  sum  of  the  whole  series  is  zero. 

Therefore,  ^— a=0,  qy  A=a. 

Removing  A— a,  it  can  be  sho^n  in  like  manner  that  B=h\ 
thence  removing  B—h^  it  can  be  shown  that  C=c ;  and  so  on. 

263.  Expansion  of  fractions  into  series. 

A  fraction  may  be  expanded  into  a  series  by  the  use  of  the 
principle  of  undetermined  coefficients. 

Example  1.    Expand  ^       ■  in  ascending  powers  of  x. 


Let  us  assume 


=A  +  Bx+Cx'  +  Dx^- 


1  +  x 

where  A,  B,  C,  etc.,  are  independent  of  x. 
Multiplying  by  1+a?, 

l  +  x'=A  +  {A  +  B)x+{B+C)x'  +  {C^-D)3(^  + 
Comparing  coefficients, 

A=l. 
A  +  B=0\  whence  ^=  —  1. 
ji5+C=l;  whence  C=2. 
O+i)=0;  whence  i)= -2. 


UNDETERMINED  COEFFICIENTS  379 


Hence,  the  required  series  is 


=l-x  +  2x^-2^  + 


1  +  x 

To  determine  what  power  of  x  shall  occur  in  the  first  term 
of  the  expansion,  we  arrange  both  numerator  and  denominator 
in  ascending  powers  of  a?,  and  perform  the  first  step  of  the 
division. 

Example  2.     Expand  »    ~o'  q  4  i^  ascending  powers  of  x. 

ZX  —  itr  —  oX 

Dividing,  the  first  term  becomes  ^xr^. 
Hence,  we  assume 

^~f^    ^=^3(r^  +  A  +  Bx-\-Cx'+Dx^+  ...... 

aX — X^  —  oX 

Multiplying  by  2iP— x"^— 3a?*, 

Comparing  coefficients, 

2A=0;  whence  A=0. 

2^—1  =  0;  whence  B=l. 

2C-A-|=-4;  whence  C=K^  +  |-4)=i. 

2D-B-^A=Q;  whence  D=^{B  +  ^A)=l. 
Hence,  the  required  series  is 

3_4a^      -1^1  +  3^+^^2+3^+ . 


2£c-ar*-3a?* 


EXERCISE    103. 


Expand  to  five  terms  in  ascending  powers  ofx: 


r=^'  l-x^'  l  +  SiC  +  a;^ 

„      l-\-X  K     £C  +  ic'  g     ^X  —  X^-^-X" 

"^^  T+^*  •    2  +  a;'  ^'  x'-x'' 


$80  ALGEBRA 

10.  1;Z^.  12.  f-^/-f..  14.       ^^+^ 


,,    l-2a;^  +  a^  ,o        1-2^'  ik     ^x'^x' 

11.  -5 — T-i — -^-  13.  .^ — ZT-S-  15. 


ar*— 4a;*— ic'^'  '  cc  +  a;^  — 2£c^*  *  1  +  a;— aj^* 

264.  Expansion  of  surds  into  series,  and  extraction  of  roots. 

The  principle  of  undetermined  coefficients  may  be  used  to 
expand  surds  into  series.  The  expansion  is  true  only  for 
those  values  of  x  which  make  the  series  convergent. 

Example  1.     Expand  yi  +  x  in  ascending  powers  of  x. 

Assume  V'l  +  x=A  +  Bx  +  Cx^  +  D,Tcr^  +  Ex*  +  Fx^+ 

Squaring,  by  the  rule  for  squaring  a  polynomial, 

l+x=A'  +  2ABx+{B'  +  2AC)x^  +  {2AD-h2BC)2(^  + 

{C'  +  2AE+2BD)x*-{-{2AF+2BE+2CD)x^+ 

Comparing  coefficients, 

A^=l;  whence  A =1. 
2AB=1;  whence  B=A^=^. 

2AC+B'=0',  whence  C=^^=-h 

2A  ^ 

2AD  +  2BC=0;  whence  D=-^^=^t^, 

2AE+2BD+C'=0-  whence  E=-^'^^±^'=-^j^. 

ZA 

.  Hence,  i/l  +  a?= 1  +  \x—\^  +  -^^x^'—^^^x"  + 


Note.— if,  in  the  above  example,  we  use  ^=-1,  different  values  for 
B,  C,  etc.,  also  will  be  obtained,  and  the  resulting  series  will  be  the 
other  square  root. 

The  same  method  may  be  used  to  find  the  square  root  of  a 
polynomial  which  is  a  perfect  square.  In  this  case  the  series 
is  finite. 


UNDETERMINED  COEFFICIENTS  381 

Example  2.     Find  the  square  root    of    4:—4:X  +  Qo(f—8a^  +  6x* 

Evidently  the  highest  and  lowest  terms  of  this  expression  must 
be  the  squares  of  the  highest  and  lowest  terms  of  the  root. 
Hence,  the  highest  term  must  contain  x^'  Accordingly,  we  as- 
sume 


V4—4x+Qx'—8x'-{-6x*—4x^+x''=A  +  Bx+Cx^+Dj(^. 
Squaring, 

4-4x+9x''—8x'  +  Qx*—4:X^  +  x^=A^  +  2ABx+{B^  +  2AC)x^ 
+  {2BC+  2AD)x'  +  {C'  +  2BD)x'  +  2CDx^  +  D'x^, 
Comparing  coefficients, 

2A5=-4; 
J52  +  2AC=9; 
2BC+2AD=-8', 
C'  +  2BD=6; 
2CD=:-4:; 
D'=l. 

Each  of  these  equations  is  satisfied  when  A =2,  B=  — 1,  C=2 
D=-l. 


Hence,  y4—'ix+9x'—8x^  +  Qx'—4x^  +  x^=2—x+2x^—x\ 

EXERCISE  104. 

Expand  to  five  terms  in  ascending  powers  of  x  : 


1.  y'l-x.  4.  i/l-^x-x\  7.  i/l-2a;. 


2.  i/l  +  9a;.  5.  yl-x  +  x\  8.   i/l+2x-x\ 


3.  i/l-4a;.  6.  i/^  +  x".  9.  i/8  +  a^. 


382  ALGEBRA 

Find  the  square  root  of : 

10.  l  +  2x+Sx'  +  2x'  +  x\ 

11.  l  +  Ax  +  lOx'-i-Ux'  +  dx'. 

12.  lQx*-'60x-nx'  +  2^x'  +  2b. 

13.  U-4x'-2ix-^Ux'-22x'-j  17x'  +  4x^. 

14.  l-4«-32a^  +  64a«-64a^  +  12a^  +  48a*. 

265.     Reversion  of  series. 

The  series  i/=A  +  I^x+  Cx^-i-Dx^+  •  •  •  •  is  said  to  be 
reverted  wlien  x  is  expressed  in  tlie  form  of  a  series  of  terms 
written  in  ascending  powers  of  y. 

A  series  may  be  reverted  by  use  of  the  principle  of  unde- 
termined coefficients. 

Example  1.     Revert  the  series  y=x+x^  +  x^  +  x*+   •  •  •  •  . 

Assume  x= Ay +  By^-{-Cy^+ Dy^+ 

Substituting, 
x=A{x+0(y'-^x^  +  x*+   •  •  ')+B{x^  +  2x^  +  Sx^+   .  .   .  .) 
+  C{x^  +  3x'+   '  ■  .)  +  D{x'+   •   •   •)• 
or,    x=Ax+{A  +  B)x^  +  {A  +  2B+C)x^  +  {A  +  3B  +  SC+D)x*+  •  •  •  •. 
Comparing  coefficients, 
A=l. 

A +1?=0;  whence B=—A=—l. 
A  +  2B+C=0;  whence  C=-A-25=l. 
A  +  3B+3O+i)=0;  whence  D=-A-^B-SC=-1. 
Hence,  x=y—y'^-\-y^—y*+  •  •  •  •  . 

EXERCISE  105. 

Revert  to  four  terms  : 

1.  y=x—x'^-{-x^—x*-i-  • 

2,  y=x  +  2x'  +  ^x'  +  4:X*-{- 


UNDETERMINED  COEFFICIENTS 


383 


^y»2  /y»3  ,-y,4 

3.  y  =  .  +  |+|-+|+    . 

5.  y=x—2x'^-^4x^—Sx*- 

rv*         ^7*2  ™3  ^y,4 

-J  rO         tl/  *o  *o 

7.  y=ic  +  £c^  +  aj''^  +  a;^+  • 


8.  y  =  a.-g-  +  ^-y- 


266.  Partial  fractions.  Two  or  more  fractions  whose  sum 
is  a  given  fraction  are  called  partial  fractions. 

The  process  of  separating  a  fraction  into  partial  fractions  is 
the  opposite  of  addition  of  fractions.  A  fraction  may  be  sep- 
arated into  partial  fractions  by  use  of  the  principle  of  undeter- 
mined coefficients. 


267.  We  consider  first  those  cases  in  which  the  numerator 
of  the  given  fraction  is  of  lower  degree  than  the  denominator. 

(1)  When  each  factor  of  the  given  denominator  is  of  the  first 
degree^  and  no  two  factors  are  equal. 

Since  the  denominator  of  the  given  fraction  must  be  the 
common  denominator  of  all  of  the  partial  fractions,  assume 
that  the  given  fraction  equals  the  sum  of  all  the  fractions 
whose  denominators  are  the  factors  of  tlie  given  denominator, 
and  whose  numerators  are  expressions  independent  of  the 
general  number  involved. 


Example  1. 


1  ft '¥•_!_  Q 

Separate  ^  ^^  \^.  into  partial  fractions. 


4x^-9 

Since  4ic2-9=(2ic  + 3) (2aj -3),  assume 
18ir  +  3        A  B 


4^_9-2iC  +  3'''2a?-3' 


334:  ALGEBRA 

Multiplying  by  4a^—9, 

■[8x  +  S=A(2x-S)  +  B{2x  +  S), 

or  i8x+S=(2A  +  2B)x  +  3B-SA. 

Comparing  coeflScients,  - 

2A  +  2B =18,  (1) 

and  SB-SA=3.  (2) 

Solving  the  system  (1),  (2),  we  get 
J.=4,  B=5. 

Hence  1?^±-^=_A_  ,  _A_. 

^®"^®'  4x^-9     2ic  +  3  +  2x-3 

Since  the  assumed  equation  is  to  be  true  for  all  finite  values 
of  the  general  number,  we  may  usually  shorten  the  work  by 
assigning  particular  values  to  the  general  number  that  will  make 
certain  undetermined  coefficients  vanish.  This  is  shown  in 
the  following  example. 

Example  2.     Separate  -5 — ^—-   into  partial  fractions. 

Since  x^—25x=x{x—5){x  +  5),  assume 

x'-75  _A       jB_       C 
x^—25x~  X    a?  +  5     X— 5* 

Multiplying  by  x^—25x, 

x^ — 75 = A(x  +  5)(a?— 5)  +  Bx{x — 5)  +  Cx{x  +  5) . 

Now  to  make  the  terms  containing  B  and  C  equal  0,  let  x=0. 

Then  — 75=— 25A;  whence  A=3. 

To  make  the  terms  in  A  and  C  equal  0^  let  x=  —  5. 

Then -50=50B;  whence  5=-l. 

To  make  the  terms  in  A  and  B  equal  0,  let  x=5. 

Then  -50=50C  ;  when  C=-l. 

Therefore,  x^-75  ^3__1___1_ 

ir— 25a?    x    x  +  5    x—5 

(2)  Wheyi  one  or  more  factors  of  the  given  denominator  are 
of  higher  degree  than  the  flrst^  and  no  two  factors  are  equal. 


UNDETERMINED  COEFFICIENTS  385 

In  this  case  evidently  the  denominators  of  the  partial  frac- 
tions will  be  the  factors  of  the  given  denominator  as  before  ; 
but  since  the  assumed  fraction  must  be  general  enough  to  in- 
clude all  fractions  with  the  given  denominator,  the  assumed 
numerator  whose  denominator  is  of  the  nth.  degree  in  the  gen- 
eral number  involved  must  be  the  most  general  expression  of 
the  degree  n—1  in  that  general  number.  Thus,  a  partial 
fraction  whose  denominator  is  x^  +  2  must  be  assumed  in  the 

form — ,. :  if  the  denominator  is  £c*  +  3,  the  numerator 

iC'^  +  S  ' 

must  be  assumed  of  the  form  Ax^ -{- Bx^ ^  Cx-\-D\  and  so  on. 
Example  1.     Separate  '   .^         into  partial  fractions. 

Since  qc^-\-1  =  {x-\-^){x'^—x-\-\),  assume 
5x^  +  1        A    .    Bx+C 


+ 


x^  +  i  ~x  4-  i     x^—x  + 1' 
Multiplying  by  o?"^  + 1 , 

^x''  +  l  =  A{x''-x^l)  +  {x  +  \){Bx-^C), 
or  ^x'  +  l  =  {A  +  B)x''  +  {B-A  +  C)x  +  A  +  C. 

Comparing  coefficients, 

A  +  B=5;  B-A  +  C=0\  A  +  C=l. 
Solving  this  system  in  A,  B,  and  C,  we  get 
A=2,  B=S,  C=-l. 

nence,    •  ^^i-^^i  +  ^^_x+l' 

(3)  When  ttco  0?'  inore /actors  of  the  gw en  denominator  are 
the  same^  i.  e.,  ichen  certain  factors  occur  to  a  poicer. 

In  this  case  a  repeated  factor  in  the  denominator  may  have 
as  many  partial  fractions  corresponding  to  it  as  the  number  of 
times  the  factor  is  repeated,  i.  e.,  as  the  power  of  the  factor. 
Their  denominator  will  be  this  factor,  raised,  respectively,  to 
\hQ  first  power,  second  power,  and  so  on,  up  to  a  power  equal  to 
25 


386 


ALGEBRA 


the  number  of  times  the  factor  is  repeated.     This  is  evident 

S  4  2 

since  any  such  fractions  as    _— ^  +  — -^2  +  T^^rjya     ^^^    ^^^ 

.,    ,    .   ,     Sx'-2x  +  l 

be   united    into  — ^ ^^-^ — • 

{x-iy 

So(^  +  Sx^ 18x 8 

Example  1.     Separate  ^^+io^ "  ^^*^  partial  fractions. 

Since  ic*  4- 43?* = orXa?  +  4) ,  assume 

8.x^  +  8x''-18a;-8_  A       BCD 
x*  +  4:3(^         ~x  +  4:    X     x^-af' 

Multiplying  by  a?* + 40?^, 

83(^  +  8x'-18x-S=Ax^  +  Bx\x  +  4:)  +  Cx{x+4:)  +  D{x+4), 
or  8o(f  +  8x'-lSx-S={A  +  B)x'  +  {4:B+C)x'  +  {4:C+D)x  +  4:D. 

Comparing  coefficients, 

A  +  B=8,4:B+C=8-  40+D=-18;  4D=-8. 
Solving  this  system  in  A,  J5,  C,  and  D,  we  get 
A=5,  B=3,  =0-4,  D=-2. 

Hence,  x'-\-4.oe'          ~x  +  4:'^x    x'    x^'  '   - 

268.  When  the  degree  of  the  numerator  is  equal  to,  or 
greater  than  that  of  the  denominator,  the  fraction  must  first 
be  reduced  to  a  mixed  expression,  then  the  fractional  part 
separated  into  partial  fractions  by  the  methods  of  §  267. 

9a?^  +  9x^ 6 

Example  1.     Separate  ..  2  ,  k^_2  ^^^^  partial  fractions. 

Reduced  to  a  mixed  expression,  by  division, 

:3X  — 2+, 


Since  3icH5.T— 2=(a?+2)(3a?— 1),  assume 

16^-10         A  B 

'dx'-\-^x—2~x+2^2>x—l' 

By  the  method  of  §267,  we  get  A=6,  B=—2. 
„  9£c'  +  9i»2-6     006  2 


UNDETERMINED  COEFFICIENTS  387 

EXERCISE    106. 

Separate  into  partial  fractions  : 

\.A, 


2£c^-cc-l  •  a3^(£cH-l) 

2  +  3a;  -                    a;''-4a;  +  3 

5  +  38a;  ..    2a;^-13a;-12 

^-  6^q^5^=^*  27-8af* 

13a;-21  ^g    2a3^  +  a;^-3a;  +  4_ 

^-  (ir-l)(a;-2)(a^  +  3)'  '  {x' -^-l^ix-X) 

^x'-\-\hx  17  6^!z:4^i. 

^-  \x'^-^:x'-x-\  '     ^X^-l)' 

7           --  +  ^-  +  1^  .               18.  Si-SI- 

'•  (a;+l)(i«  +  2)(a:+3)  a, +a; +i 

■        1  19    -^  +  2«'  +  5 

8.  ^j^-  ^*''  (a?-l)(a;+l) 

„     Sa^'-ia  20      2a;^-g«^-^ 

^-  (2x-3)»-  (a:»  +  2)(x'  +  2) 

10. .  r^ ,..  •    21-  "'"' 


23. 
24. 


8  +  12a;-2a;^-14a;=^-10a;*-2ar^ 

■  x\x+2y 


CHAPTER  XXVI, 
LOGARITHMS. 

269.  Exponential  equations.  In  all  the  equations  which  we 
have  discussed,  the  unknown  numbers  have  appeared  as  bases, 
with  known  coefficients  and  exponents.  There  are  problems 
which  lead  to  equations  in  which  the  unknown  numbers  ap- 
pear as  exponents.  Such  equations  are  called  exponential 
equations. 

Thus,  2^=16  is  an  exponential  equation,  the  unknown  number 

appearing  as  an  exponent.     The  solution  of  this  equation  is  a? =4. 

4^=8  is  an  exponential  equation.     Its  solution  is  ir=|,  because 

4^=V?=8. 

270.  The  solutions  of  some  exponential  equations  can  be 
found  easily  by  inspection.     But  in  general  this  is  impossible. 

Thus,  to  solve  3^=243  is  to  find  the  power  to  which  3  must  be 
raised  to  give  243.     This  is  seen  by  trial  to  be  5.     Hence,  x=5. 

But  2^=12  cannot  be  solved  by  inspection,  x  is  more  than  3, 
because  2-^=8;  and  x  is  less  than  4,  because  2*=16.  Hence, 
x=3+a  fraction.  In  fact,  x  here  is  what  is  called  an  incommen- 
surable number  whose  exact  value  cannot  be  found. 

The  general  exponential  equation  can  best  be  solved  by  the 
aid  of  a  set  of  numbers  called  logarithms. 

271.  Logarithms.     T\\q  logarithm  of  a  number  i'^  the  expo 
nent  which  indicates  the  power  to  which  a  given  base  must  be 

388 


LOGARITHMS  389 

raised  to  produce  that  number.     That  is,  if  a*=w,  then  x  is 
the  logarithm  of  n  to  the  base  a  ;  and  is  written 


Thus,  since  2=^=8,  log28=3. 


Since  3*=81,  log381=4.  j-v  r  '-  --£■  ^^'='  -  ^-? 

Since  5*= 625,  log5625=4.  '>*-^  ^-  -^  ^  ^' 


Since  4-2=i:,^-~,  log4T'^=~2. 

Since  9-^=~|=.^V,log9^V=-|.        ^x^.^;^- ~%, 

/       It   follows   that   the  exponential   equation   a^  =  n   and   the 
logarithmic  equation  aj=log,,?i  are  equivalent. 


EXERCISE  107. 

Express  the  followijig  relations  in  terms  of  logarithms 

1.  2^=^32.  ^  32-sr3.  7^=343.  5.  5«  =  15625. 

2.  3*  =  81.     '^  4.  10^  =  10000.  6.  4-'=gL-. 


7.  5-2 =J-^.  8.  10 


2_       1 


2T-  ^'^    -^^      —  TO  0-  ^  -^f^ 


Vo 


I  60 


^Express  the  following  relations  by  means  of  exponents  : 

9.  log39-2.  "3'i;^    11.  log,16  =  2.  13.  log,^V=-2.     Z'^:^-^ 

10.  log2l6=4.  12.  logs4=f.  14.  logio.001  =  -3. 

15.  log,oo.001  =  -f. 

Find  the  values  of  the  following  logarithms: 

16.  logaS.  20.  log,64.  24.  log6216. 

17.  log,327.  21.  log,  .5.  25.  logsyi^. 

18.  logiolOO.  22.  log2.25.  26.  log^ol. 

19.  logio-OOOl.  23.  logger  27.  log.l. 


390  ALGEBRA 

To  the  base  4  what  numbers  have  the  following  logarithms  ? 

28.  1.     U  30.  f  32.   -i. 

29.  3.  31.  -2.  33.  5. 

34.  -4. 

272.  Fuiidamental  principles  of  logarithms. 

Fi'om  the  definition  of  a  logarithm  it  follows  that  any  posi- 
tive number,  except  1,  may  be  used  as  the  base  of  the  logarithm 
of  any  arithmetical  number.  The  following  fundamental 
principles  apply  to  logarithms  to  any  base. 

(A)    The  logarithm  of  1  to  any  base  is  0  ;  that  is, 

This  principle  follows  from  «"=1. 

{£)    The  logarithm  of  the  base  itself  is  1 ;  that  is, 

%„a  =  l. 
This  follows  from  a^  =  a. 

(  C)  The  logarithm  of  a  product  equals  the  sum  of  the  loga- 
rithms of  Its  factors  ;  that  is, 

( loa^mn  =  loq.m  +  loa^n.     \^ 

To  pjrove  this,  let  a^=m^  and  ay  =  n.  \ 

Then  m)i  =  a''-ay=a^+y. 

Hence,  lQgam?i=a;  +  y  =  log„m  +  log„?i. 

In  like  manner,  this  principle  can  be  proved  to  hold  for  any 
number  of  factors. 

{!>)  The  logarithm  of  a  quotient  equals  the  logarithm  of  the 
dividend  minus  the  logarithm  of  the  divisor;  that  is, 

l^9a{z]  =logam—log^n. 
To  prove  this,  let  «^=m,  and  a«'=n. 
Then  -.=a^-^a?'=Q-c-y^ 


LOGARITHMS  391 

Hence,  log„  (^J  =a;-y =log„m-log„w. 

(^)  The  logarithm  of  a  poicer  of  a  number  equals  the  loga- 
rithm of  the  number,  multiplied  by  the  exponent  of  the  power; 
that  is, 

To  prove  this,  let aJ^^n.  — 

Then  7i^ ={a''Y= ap*. 

Hence,  \og^{n^=px=p  log^^w. 

{F)  The  logarithm  of  a  root  of  a  number  equals  the  logarithm 
of  the  nwnber^  divided  by  the  index  of  the  root;  that  is, 

To  prove  this,  let 

Then  l/^=l/a^=a''.     - 

Hence,  ,  log„i/  7i=~  -^ ,  or  -  log„;i. 

Note.  —Principle  {F)  might  be  considered  a  special  case  of  {E) , 

yn  being  written  as  the  power  n^. 

By  the  use  of  the  above  principles  we  shall  be  able  to  replace 
the  operations  of  multiplication  and  division  by  those  of  addi- 
tion and  subtraction,  and  the  operations  of  involution  and 
evolution  by  those  of  multiplication  and  division. 

Example  1.  Express  loga"^  in  terms  of  logaO?,  loga?/, 
log„2;,  and  lo^a^v. 

\og~~=  =\<^axy^-^o^aZ\/'^  By  (D). 

Zy   tV       / 

=l0gaa?  +  l0ga2/'-l0ga2;-l0gal/w  By  (0). 

=logaa;  +  2  loga2/-loga2;-i  log„?<^.      By  {E)  and  {F), 


392  ALGEBRA 

Example  2.     Express  2  logaic— |  logaV  +  h  ^^SaZ  as  a  single  log- 
arithm. 
2  logaX-f  loga2/  +  i  log„2;=log„ar^-log„2/^  +  loga2^^    By  (^). 

=log„^  By(C)and(i)). 

Examples.     If  logio2=.3010,  logio3=.4771,   find  logio|/6T 

logio|/6=^  Iogio6=|(logio2  +  logio8) 
=^(.3010  +  . 4771) 
=  .3890. 
Example  4.     If     logio2=.3010,     logio3=.4771,      logio5=.6990, 

3y 

find     log,o^^. 

1/800 

Factoring,  360=23-3='-5;  800=2^-5^ 

I/' 360  3 . 

'    ^^^i«^*^^=logioi/360-logioT/800 

=ilog,o2^-3=^-5-ilogio2^-5' 
= K3  logio2  +  2  logio3  +  logioS)  - 

I(5logio2  +  2logio5) 
=K.9030  +  .9542  +  .6990)-i(1.5050  +  1.3980) 
=.1263. 

EXERCISE  108. 

Express  the  following  logarithms  in  terms  of  log„£c,  log.y, 
log„2j,  log„^  .• 


''^-  ^  6.  log^-!^. 

2.  log^a^y.  /_   ^_  1/^1/^ 

K    1       1/^1/ y  a; 


LOGARITHMS  893 

B.log.g.  10.1ogA-*  ^^  ^ 

9   loff  (^^V.  Ill  -"  «5~* 


2a;y'' 


13.  l„g„/^+logy_^.  14.  log„3^^^ 

Express  as  single  logarithms  : 

15.  log„a5+log„y  +  log„2-log„?^. 

16.  log„aj-log„y  +  log„^-log„^. 

17.  2  log^a;  +  2  log„y-2  log„2-2  log„«^. 

18.  i  log„a;-i  log„y. 

19.  }log„^()  +  |lbg„2. 

20.  3  1og„a;-ilog„(y  +  ^). 

21.  3log„g)+2  1og.(| 

If  logio2  =  .3010,   logio3  =  . 4771,    logio5  =  . 6990,    logio7  =  .8451, 
find  to  the  base  10  the  logarithms  of  the  following  numbers : 

22.  18.  29.  70.  l/98 

3d.  ■  ., 

23.  15.  30.  210.  y  144 

24.  20.  31.  1000.     .  _  3  ^  • 

25.  50.  32.  6|.  37.  ^;^l'g. 

26.  24.  33.  i/rs.  1^5  1  7 

27.  21.  34.  yW^  ^      (5^3 

28.  If.  35.  1^180.  ''°-  (5|)^- 

273.  Common  logarithms.     The   logarithm  of  a  number  to 
the  base  10  is  called  a  common  logarithm. 

Thus,  logio6,   \og^f,124:,  logjo  t\j   logjo-lOOO,    are  common- loga- 
rithms. 


394:  ALGEBRA 

The  logarithms  of  all  arithmetical  numbers  to  any  one  base 
constitute  a  system  of  logarithms.  The  common  logarithms  of 
all  arithmetical  numbers  constitute  what  is  called  the  common 
system  of  logarithms.*  The  common  system  of  logarithms  is 
used  in  practical  calculations.  This  system  is  superior  to 
other  systems  for  practical  use  because  its  base  10  is  also  the 
base  of  our  decimal  system  of  numbers. 

The  rest  of  the  discussion  in  this  chapter  will  be  confined  to 
commo?i  logarithms ;  and  in  the  following  sections  the  base 
will  be  understood  to  be  10,  and  will  not  be  written. 

274.  Characteristic  and  mantissa. 

From  the  definition  of  a  logarithm  we  have  : 

since  10"  =  1,  log  l'=0  ; 

since  10^  =  10,  log  10  =  1; 

since  10=^  =  100,  log  100  =  2; 

since  10=^=1000,  log  1000  =  3;  etc.; 

and 

since  10-^  =  .l,  log  .1  =  — 1 ; 

since  10-^  =  .01,  log  .01  =  -2; 

since  10-='  =  .001,  log  .001  = -3  ;  etc. 
It  Is  evident  from  the  above  that  the  logarithm  of  a  positive 
integral    power   of   10   is   a   positive   integer,   and   that    the 

*  The  system  of  common  logarithms  was  introduced  in  the  seven- 
teenth century  by  Henry  Briggs.  Accordingly,  logarithms  to  the  base 
10  are  also  called  Briggs*  logarithms.  Another  important  system  is 
the  Napierian  system,  named  after  Napier,  a  contemporary  of  Briggs. 
Tlie  base. of  the  Napierian  system  is  the  sum  of  the  infinite  series 
.  ,  1     1111 

fI~'~J2'*"p'^|4'^J5 •     "^^'^  s""^  ^^  this  series  is  approximately 

2.7182818,  and  is  represented  by  the  letter  e.    Napier  himself,  however, 
did  not  use  the  base  e  in  his  system. 

While  the  common  system  is  used  in  practical  calculations,  the 
Napierian  system  is  used  in  theoretical  investigations. 


LOGARITHMS  395 

logarithm  of  a  negative  integral  power  of  10  is  a  negative 
integer. 

Moreover,  since  65  is  greater  than  10^  and  less  than  10^  log 
65  will  be  greater  than  1  and  less  than  2.  Hence,  log  65  =  l  +  rt 
decimal.  Also,  since  382  is  greater  than  10^  and  less  than  10^, 
log  382  will  be  greater  than  two  and  less  than  3.  Hence,  log 
Z%'±  =  1-\r  a  decimal 

Evidently,  the  logarithm  of  any  positive  number,  except  a 
positive  or  negative  integral  power  of  10,  will  consist  of  an 
integral  ami  a  decimal  part. 

Thus,  log  825=2.9165,  to  four  decimal  places.* 

The  integral  part  of  a  logarithm  is  called  its  characteristic,  and 
the  decimal  part  is  called  its  mantissa. 

275.  Determination  of  tlie  characteristic 

A  number  having  one  figure  in  its  integral  part  lies  between 
10"  and  10\  Hence,  its  logarithm  lies  between  0  and  1 ;  i.  e., 
it  equals  0  +  «  decimal.  A  number  having  two  figures  in  its 
integral  part  lies  between  10^  and  101  Hence,  its  logarithm 
lies  between  1  and  2  ;  i.  6.,  it  equals  1  +  a  decimal.  Similarly, 
if  a  number  has  three  figures  in  its  integral  part,  its  logarithm 
lies  between  2  and  3;  i.  e.,  it  equals  24-«  decimal^  and  so  on. 

Therefore,  if  a  number  is  greater  than  i,  the  characteristic  of 
its  logarithm  is  positive,  and  is  less  by  1  than  the  number  of 
figures  in  the  integral  ]Kirt  of  the  number. 

Thus,  in  log  6841.27  the  characteristic  is  3.  In  log  362.781  the 
characteristic  is  2.  . 

Again,  a  number  less  than  1,  and    having  no  zero  imme- 


*  It  should  be  remembered  that  the  decimal  part  is  an  incommen- 
surable number,  and  may  carried  to  any  degree  of  accuracy  required. 


396  ALGEBRA 

diately  following  the  decimal  point,  lies  between  lO""  and  10-^ 
Hence,  its  logarithm  lies  between  0  and— 1 ;  i.  e.,  it  equals 
— 1  +  «  decimal  A  number  less  than  1,  and  having  one  zero 
immediately  following  the  decimal  point,  lies  between  10"^ 
and  lO-l  Hence  its  logarithm  lies  between  —  1  and  —  2  ;  ^.  e.,  it 
equals  — 2 +  «<^ecma^.  Similarly,  if  a  number  less  than  1 
has  two  zeros  immediately  following  the  decimal  point,  its 
logarithm  will  lie  between  —2  and  —  3  ;  i.  e.,  it  equals  —  3-f-a 
decimal ;  and  so  on. 

Therefore,  if  a  number  is  less  than  i,  the  characteristic  of  its 
logarithm  is  negative^  and  is  greater  by  1  than  the  number  of 
zeros  immediately  following  the  decimal  point. 

Thus,  in  log  .0683  the  characteristic  is  —2.  In  log. 00.08  the 
characteristic  is  —4.     In  log  .3974  the  characteristic  is  —1. 

In  writing  the  logarithm  of  a  number  the  mantissa  is  always 
positive,  and  if  the  characteristic  is  negative,  the  minus  sign 
is  written  above  the  characteristic  to  signify  that  it  applies  to 
the  characteristic  alone. 

Tl^s,  log  .00357=3.5527.     This  means  -3  +  .5527. 

276.  IVie  common  logarithms  of  7iumbers  which  do  not  differ^ 
except  in  the  position  of  the  decimal  pointy  have  the  same 
mantissa. 

This  follows  from  the  fact  that  changing  the  position  of  the 
decimal  point  in  a  number  is  equivalent  to  multiplying  or 
dividing  the  number  by  some  integral  power  of  10. 

Thus,  3.261  X  10=32.61;  3.261  x  102=326.1 ;     3.261--102=. 03261. 

But  when  a  number  is  multiplied  or  divided  by  a  power  of 
10,  an  integer  is  added  to,  or  subtracted  from,  its  logarithm; 
hence  the  mantissa  is  not  changed. 


LOGARITHMS  "  397 

For  example,  it  has  been  found  that 
log  2680=3.4281; 
lience,  log  268  =2.4281; 

log  26.8=1.4281; 
log  2.68  =  . 4281; 
log  .268=1.4281. 

277.  Tables  of  Mantissas.  The  common  logarithms  of  sets  of 
consecutive  integers  have  been  computed  and  tabulated.  At 
the  end  of  this  chapter  is  a  table  which  contains  the  mantissas 
of  the  logarithms  of  all  integers  from  1  to  1000. 

Since  the  mantissa  of  the  logarithm  of  a  number  depends 
only  upon  the  sequence  of  figures,  and  not  upon  the  position  of 
the  decimal  point,  only  the  mantissas  of  the  logarithms  of  in- 
tegers need  be  tabulated.  Since  the  characteristics  may  be 
found  by  the  rules  of  §  276,  they  are  left  out  of  the  table. 

Logarithms  may  be  computed  to  any  number  of  decimal 
places,  the  number  depending  upon  the  degree  of  accuracy  re- 
quired in  their  use.  In  the  table  in  this  book  the  manttesas 
are  computed  to  four  decimal  places.  In  this  table  the  first 
two  figures  of  each  number  are  found  in  the  column  headed  JV, 
and  the  third  figure  in  the  horizontal  line  at  the  top  of  the 
table.  The  mantissas,  with  decimal  points  omitted,  are  found 
in  the  columns  headed  0,  1,  2,  3,  etc. 

In  finding  the  logarithm  of  a  number,  find  its  characteristic 
by  §  275,  and  look  in  the  tal)le  for  the  mantissa.  In  looking- 
for  the  mantissa  of  a  number  containing  less  than  three  figures, 
annex  ciphers  until  it  has  three  figures. 

Thus,  to  find  the  mantissa  log  38,  look  for  the  mantissa  of  380. 
To  find  the  mantissa  of  log  3,  look  for  the  mantissa  of  log  300. 


398  ALGEBRA 

278.  Use  of  the  table ;  to  find  the  logarithm  of  a  given  number. 

{a)  When  the  given  number  contains  not  more  than  three 
figures,  the  mantissa  of  its  logarithm  is  obtained  directly  from 
the  table. 

Example  1.    Find  log  32.7. 

By  §  275  its  characteristic  is  1.  By  §  276  the  required  mantissa 
is  the  mantissa  of  log  327. 

Look  for  32  in  the  column  headed  N.     Looking  along  the  hori- 
zontal line  of  numbers  opposite  32,  to  the  column  headed  7,  we 
find  .5145,  the  required  mantissa.    Hence, 
log  32.7=1.5145. 

Example  2.    Find  log  .91. 

By  §  275  the  characteristic  is  —1.  The  required  mantissa  is 
the  mantissa  of  log  910,  This  is  opposite  91  in  the  column  headed 
0,  and  is  seen  to  be  .9590.     Hence, 

log  .91=1.9590. 

{h)  When  the  given  number  contains  more  than  three  figures, 
use  is  made  of  the  principle  that  when  the  difference  of  two 
numbers  is  small  compared  with  either  of  them,  the  difference 
of  the  numbers  is  approximately  proportional  to  the  difference 
of  their  logarithms. 

Example  3.    Find  log  2.8465. 

Shift  the  decimal  point  until  it  follows  the  third  figure. 

The  required  mantissa  is  that  of  log  284,65. 

Now  284,65  is  greater  than  284  by  .65. 

The  mantissa  of  log  284=, 4533. 
,    The  mantissa  of  log  285 =,4548. 

Subtracting,  .4548— ,4533=. 0015. 

Hence,  an  increase  of  1  in  284  causes  an  increase  of  ,0015  in 
the  corresponding  mantissa.  Therefore,  an  increase  of  .65  will 
cause  an  increase  of  .65  X -0015,  or  approximately  .0010,  in  the 
mantissa. 


LOGARITHMS 

Adding  .0010  to  the  mantissa  of  log  284  gives  .4543. 
Attaching  the  characteristic,  we  have 
log  2.8465=0.4543. 

Example  4.     Find  log  .008214. 

The  characteristic  is  —3. 

The  required  mantissa  is  the  mantissa  of  log  821.4. 

Mantissa  of  log  821^.9143. 

Mantissa  of  log  822 =.9 149. 

.9149-.9143=.0006. 
.4X.0006  =  .0002. 

.9143  +  . 0002=. 9145. 
Hence,  log  .008214=3.9145. 


S99 


EXERCISE  109. 


Find  the  logarithms  of: 


1.  215. 

11. 

100. 

21.  .06843. 

2.  673. 

12. 

900. 

22.  4268.4. 

3.  940. 

13. 

1. 

23.  1.096. 

4.  717. 

14. 

2. 

24.  .00012. 

5.  4,62. 

15. 

1684. 

25.  99.99. 

6.  19.9. 

16. 

34.27. 

26.  2031.7. 

7.  830. 

17. 

100.5. 

27.  .0083326 

8.  16. 

18. 

926.81. 

28.  ,50416. 

9.  29. 

19. 

.8632. 

29.  68593. 

10.  8.5. 

20. 

.00315. 

30.  .074803. 

279.  To  find  a  number  whose  logarithm  is  given. 

(«)  When  the  given  mantissa  is  found  in  the  table^  the 
sequence  of  figures  of  the  required  number  may  be  obtained 
by  reversing  the  process  (a)  of  §  278.  The  position  of  the 
decimal  point  is  determined  by  reversing  the  rules  in  §  275. 


400  ALGEBRA 

Example  1.     Find  the  number  whose  logarithm  is  1.9325. 
Looking  in  the  table,  we  find  the  mantissa  .9325  opposite  85 
and  in  the  column  headed  6.     Hence, 

.9325= the  mantissa  of  log  856. 
Since  the  characteristic  is  1,  the  number  must   contain  two 
figures  to  the  left  of  the  decimal  point.     Hence, 
1.9325=log  85.6. 

Example  2.    Find  the  number  whose  logarithm  is  2.5289. 
From  the  table  we  have 

.5289=mantissa  of  log  338. 
Since  the  characteristic  is  —2,  the  number  must  be  less  than  1, 
and  must  contain  one  zero  immediately  to  the  right  of  the  decimal 
point.     Hence, 

2.5289=log  .0338. 

(b)  'When  the  given  mantissa  is  not  found  in  the  table^  the 
required  number  is  obtained  by  reversing  the  process  of  (i), 
§  278. 

Example  3.     Find  the  number  whose  logarithm  is  3.6496. 

The  mantissa  .6496  is  not  in  the  table,  but  the  mantissas  of  the 
table  between  which  it  lies  in  value  are  .6493  and  .6503. 

Now  .6496  is  greater  than  .6493  by  .0003. 

Also  .6493= mantissa  of  log  446, 
and       .6 503= mantissa  of  log  447. 

Subtracting,  .6503-. 6493=. 0010. 

Hence,  an  increase  of  .0010  in  the  mantissa  .6493  causes  an  in- 
crease of  1  in  the  corresponding  number.  Therefore,  an  increase 
of  .0003  will  cause  an  increase  of  .0003-^.0010,  or  .3,  in  the 
number. 

Adding  .3  to  446  gives  446.3. 

Therefore,  .6496=mantissa  of  log  446.3. 

Since  the  characteristic  is  3,  we  have 
3.6496=log4463. 


LOGARITHMS  401 

Example  4.     Find  the  number  whose  logarithm  is  ^3.8684. 
This  mantissa  is  not  in  the  table,  but  the  next  less  mantissa  is 
.8681,  and  the  next  greater  is  .8686. 

Now  .8681=mantissa  of  log  738, 

and  .8686= mantissa  of  log  739. 

.8686-. 8681  =  . 0005, 
and        .8684-. 8681  =  . 0003. 
.0003-^. 0005  =  . 6. 
■  738  +  . 6=738.6. 
Hence,  .8684= mantissa  of  log  738.6. 

Locating  the  decimal  point,  we  get 

3.8684=log  .007386. 

EXERCISE    110. 

Find  the  numbers  whose  logarithms  are : 


1. 

1.7582. 

6. 

6.6884. 

11. 

4.0096. 

16. 

5.0220. 

2. 

3.8615. 

7. 

2.4786. 

12. 

1.4703. 

17. 

3.0392. 

3. 

.1847. 

8. 

3.6021. 

13. 

2.9765. 

18. 

4.4756. 

4. 

T.4609. 

9. 

3.7251. 

14. 

2.8460. 

19. 

.8735. 

5. 

2.6804. 

10. 

2.8976."^ 

15. 

1.4072. 

20. 

1.5734. 

280.  Cologarithms.  The  cologarithm  of  ic  is  the  logarithm  of  — 
From  this  definition  it  follows  that 

colog  x=/og(-J  =log  l-/og  x=—log  x; 

that  is,  the  cologarithm  of  a  number  may  he  obtained  by  changing 
the  sign  of  its  logarithm. 

Since  the  mantissa  of  a  logarithm  is  alwaj^'s  written  positive, 
to  change  the  sign  of  the  logarithm  would  give  a  negative  man- 
tissa.    In  order  to  avoid  a  negative  mantissa  in  the  cologarithm, 
26 


402  ALGEBRA 

usually  in  place  of  —log  £c,  its  equivalent,  (10— log  cc)  — 10,  is 
used.  Evidently,  therefore,  the  cologarithm  of  a  number  may 
be  found  by  subtracting  the  logarithm  of  the  number  from  10, 
and  indicating  the  addition  of  —10  to  the  remainder. 

Example  1.     Find  colog  485. 

log  485=2.6857. 
Hence,  colog  485=(10-2.6857)-10=7.3143-10. 

If  log  X  lies  in  absolute  value  between  10  and  20,  then  in 
order  to  make  the  mantissa  positive  we  use  for  colog  x  the 
form 

(20-log£c)-20. 

In  general,  if  convenient,  we  may  use  the  cologarithm  in  the 
form 

{a— log  x)— a, 

where  a  is  any  number  that  will  make  the  mantissa  positive. 

Example  2.     Find  colog  267000000000.    ' 

log  267000000000=11.4265. 
Hence,  colog 267000000000=(20-11.4265)~20=8.5735-20. 

If  the  characteristic  of  the  logarithm  is  negative,  then  the 
—  10,  or  —20,  will  disappear  from  the  value  of  the  cologarithm. 

Example  3.    Find  colog  .00814. 

log  .00814=3.9106. 
Hence,  colog  .00814= (10-3.9106) -10, 

=  (10  +  3-.9106)-10, 

=12.0894-10, 

=2.0894. 

It  has  been  shown  that  to  obtain  the  logarithm  of  a  quotient, 
we  subtract  the  logarithm  of  the  divisor  from  the  logarithm 
of  the  dividend.  Since  colog  x=  —log  «,  instead  of  subtracting 
the  logarithm  of  the  divisor  ice  may  add  its  cologarithm. 


Example  4.     Find  log 

1        6827 
^^^  8iT6 


Hence, 


LOGARITHMS 

6827 
81.6' 

log  6827 +  colog  81.6. 


log  6827=  3.8342. 
colog81.6=  8.0883-10. 

log  ^=11.9225-10, 
=  1.9225. 


403 


EXERCISE  111. 

Find  the  cologarithm  of  : 

1.  72.8.  6.  68.27.  11.  .00321. 

2.  691.  7.  1375.  12.  .6847. 

3.  4.56.  8.  261.3.  13.  .0315. 

4.  326.7.  9.  18329.  14.  .0000623. 

5.  12.34.  10.  43165.  15.  .0004721. 

16.  .0005638. 

281.  Computation  by  logarithms.  By  the  use  of  logarithms 
long  computations  can  be  avoided.  Multiplication^  division, 
involution,  and  evolution,  may  be  replaced  by  addition,  sub- 
traction, midtiplication,  and  division,  respectively. 

In  using  logarithms  with  negative  characteristics  it  is  some- 
times convenient  to  add  some  number  to  the  logarithms,  in 
order  to  make  the  characteristics  positive,  then  to  indicate  the 
subtraction  of  the  number.     Thus,  3  .6271=^7.6271-10. 

Example  1.     Find  |^ ."00327. 

We  first  find  log  i^. 00327. 

log  1^.00327  =i  log  .00327  §  272,  F. 

=i  (3.5145) 
=^  (2.5145-5) 
=^5029-1 
=1.5029. 


40^ 


ALGEBRA 


]Sl4>w  1.5029=log  .318555. 


621.3X.03247 
71.8 


Hence,  x7.00327=. 318555. 
Example  2.  Find  the  value  of 
Taking  the  logarithm, 

^^^621.3X.03247_^^^  621.3  +  log.03247  +  colog  71.8. 
71.8 
log  621.3=2.7933, 

log. 03247 =2.5115=8.5115-10, 

colog  71.8=8.1439-10, 

Sum=19.4487-20 

^    =1.4487. 

But  1.4487=log  .281. 

rru — ^e^^^   621.3X03^47 ^q^    „^r^l„^^ir^n^■a^■^ 


xj.x\:/iCJLvi.c,  

"     71    8                          .«^*,    «^|^a.v^^x*xxo*wv.u.j  . 

Example  3. 

Find  the  sixth  power  of  .428. 

log  .428«=6  log  .428 

=6(1.6314) 

=  6(9.6314-10) 

=  57.7884-60 

• 

=3.7884. 

But 

3.7884=log  .00614. 

Therefore, 

.428«=.00614,  approximately. 

Example  4. 

Find  the  value  of  2.476  X  (-1.724). 

The  sign  of 

the   product  must  be  determined  by  the  laws  of 

signs.     By  logarithms  we  obtain  merely  the  absolute  value  of  the 

product. 

. 

We  have 

log  2. 476 =.3938 

log  1.742=. 2410 

.6348 

Now 

.6348=log  4.313. 

Therefore, 

2.476 X (-1.742)  =  -4.313,  approximately. 

LOGARITHMS  405  > 

5/- ^  ^         ^^H^f-J-y 

Example  5.     Find  the  value  of  ^  /     •  Q27^ X 32. 6 X |/ 54^«-j:„ ,  ,or 


/     .Q27^X32.6Xt/54Kc^.    or        '  •' 


Eepresent  this  expression  by  x. 

Then  log  a?=i(4  log  .027  +  log  32.6  +  1  log  542  + 1^  colog  4.12 
+  1  colog  7.14+^  colog  6.28) 

4  log  .027=7.7256=3.7256-10 

log  32.6=1.5132 

i  log  542=   .6835 

i  colog  4.12=^29. 3851-30)  =  9. 7950-10 

i  colog  7.14=K49.1463-50)  =  9.8292-10 

^  colog  6. 28=i(59.2020-60)  =  9. 8670-10 


loga?= 

5)  35.4135-40 

7.0827-8 

= 

1.0827. 

But 

r.0827z 

=log  .12097. 

Therefore, 

x= 

=.12097,  approximately. 

EXERCISE  112. 

Find  by  logarithm 

s  the  value  of : 

1. 

71.6X.327. 

8. 

.1965--18.97. 

2. 

1.068  X. 0039. 

9. 

6.765-f-(-.01286) 

3. 
4. 
5. 
6. 

681.7X4.235. 
3.1416X1728. 
•1.414x(--0632). 
(-4617)X(-.03269). 

10. 
11. 

12. 

5334X.02374 
-47.43X3.246* 

2.476x73.81 
.524x6184* 
1657 

7. 

7.631--6214. 

1.025X326.81* 

13. 

17.86^ 

16.  219.4^ 

17.  V  9.268". 

14. 

.06814'". 

16.  (- 

-.2596)*. 

18.  i>6278.i: 

406  ALGEBRA 

19.  i> -20035.  21.  .684*.  ^^-  ('^A)*- 

20.  .068^  22.  (^)«.  24.  3.69i-^. 

25.  (.6827X.114)^ 


26. 


y       27*  X -028^X16.75^ 
y     i/629Xi>87I0Xt5/12C3* 


5    / 


27.  98.7i/  .068X1/28.59 

'  ^     i^6Xi>206:4Xi/:009l* 
282.  The  solutions  of  exponential  equations.     The    solution 
of  an  exponential  equation  may  be  obtained  by  the  aid  of 
logarithms. 

Example  1.     Solve  52^-11(5^) +24=0. 
Factoring,  (5^-8)(5^-3)=0. 

■   Hence,  5^=8,  (1) 

or  5^=3.  (2) 

From  (1),  taking  the  logarithm  of  both  members, 
X  log  5=log  8. 

Therefore  *       a,-l2g_§_:i9031_ 

±nererore,  ^~log  5~.6990--^''^^^^- 

From  (2),  a;log5=log3. 

Therefore  r-^-^^-^i^-  882^ 

ineretore,  x-^^^  .--^gg^-.6825. 

Example  2.     Solve  5*^-^=4^-1. 

Taking  the  logarithms  of  both  members, 

(3^-l)log5=(.r-l)log4. 
Removing  parentheses, 

Sx  log  5— log  5=x  log  4— log  4, 
or  ar(3  log  5 -log  4)= log  5— log  4. 

Hence,  ^^  Jog  5 -log  4 

3  log  5-log  4 
_  .6990-. 6021 
■"2.0970 -.6021 
=0648. 


LOGARITHMS  407 

EXERCISE  113. 

Solve  : 

1.  3^=81.  6.  23^=25.  9.  4^+1  =  8-2^+2. 

2.  5'--25  =  0.  6.  4^-1  =  3^+1.  10.  '\/'2^^=i/¥=^. 

3.  2-^  =  64.  7.  2^-1  =  .32^-5.  11.  1/3^1=^ =3»-^. 

4.  3^=15.  8.  52^—12^+1=0.  12.  2'««'^=8. 


13.  5'«'^^  =  625.  14.  176.82  =  2.36. 

15.  32-^-4(3-)-12  =  0. 

16.  2*^-3(22-^)  =4. 

17.  2(4^)-4^-6-0. 

EXERCISES   FOR  REVIEW  (VIII). 

1.  Find  the  value  of  a  in 

a+2     4-ffl_7 
a  —  1       2a       3' 

2.  Find  the  value  of  t  in 

16 


l/«^-l  +  6=; 


3.  Solvea!(£c''-4)  +  (a3-2)=0.    .  . 

4.  Solve  for  x  and  y 

^^^  +  y^-*- 

5.  Solve  (a;-l)(£c  +  2)(aj^-6ic  +  9)=0. 

6.  On  how  many  nights  may  a  different  guard  of  5  men  be 
taken  from  a  body  of  26  men  ? 

1\2« 

7.  What  term  in  (x-] —  j     does  not  contain  x  ? 

8.  What  is  'dn  undetermined  coefficient  ? 


408  ALGEBRA 

9.  What  is  the  theorem  of  undetermined  coefficients  ?     Upon 
what  principles  did  the  proof  in  this  book  depend  ? 

10.  What  use  can  be  made  of  the  theorem  of  undetermined 
coefficients  ? 

Make  use  of  the  method  of  undetermined  coefficients  in  the 
following  : 

11.  Expand    ..  _o    to  five  terms  in  ascending  powers  of  x. 

12.  Expand    ^_   _  i  mto  a  series. 

\—x 

13.  Expand    ..  .      ,    ^  into  a  series. 


14.  Expand  j/l—Sx  to  five  terms  in  ascending  powers  of  x, 
in  two  Avays. 

15.  What  is  reversion  of  series  ?  Revert  to  four  terms 
y=x—Sx^-i-bx^—7x*+  •  •  •  . 

16.  What  are  partial  fractions  ? 

17.  In  what  way  can  a  given  fraction  be  separated  into 
partial  fractions  ? 

18.  In  what  form  must  the  partial  fractions  be  assumed 
when  each  factor  of  the  -given  denominator  is  of  the  first 
degree,  and  two  or  more  of  the  factors  are  equal  ?    Illustrate. 

19.  In  what  form  must  numerators  be  assumed  when  a 
factor  of  the  given  denominator  is  of  higher  degree  than  the 
first,  and  no  two  factors  are  equal  ?     Why  is  this  ? 

20.  Separate  into  partial  fractions  : 

(n\        ^  +  ^^-^'  r^^     2^  +  2 

^""^  {i-x){i-^xy'  (^)  ^(^17* 


LOGARITHMS  409 

21.  What  is  an  exponential  equation  ?     Illustrate. 

22.  Solve  2^  =  128;  4-  =  G4;  9'^  =  27. 

23.  Give  the  integers  between   which  the  solutions  of  the 
following  equations  lie :  2^  =  3;  7-^  =  100;  5^  =  600. 

24.  How   can   the   solutions    of   exponential    equations    be 
obtained  when  they  can  not  be  seen  by  inspection? 

25.  What  is  a  logarithm 'i 

26.  To  what  exponential  equation  is  logio65  =  i:c  equivalent? 

27.  Find   the    value   of  10^^49 ;   log g 216;   loggyV^   ^^^a^^  ^ 
log„a^;  log  J;  log^a. 

28.  What  is  a  system  of  logarithms  ? 

29.  What  are  common  logarithms  ? 

30.  Give     in     terms     of     log^aj     and     log„y,   {a)    log^icy ; 

ih)  iog„(^^)  ;  (c)  logx;  (^0  log^r  X. 

31.  Prove  the  following  identities  : 

(«)  log,.(m-f-/i)  —\og^m—\o^^n ; 
{h)  log6mP=^logftm;  (c)  logioeXlog,10  =  l. 

32.  Give  the  equivalents  of  the  following  and  show  how  you 
get  them  : 

{a)  log„a;  {h)  logj;  (c)  lcg„(l-~a)  ; 

{d)  log,o.001;  (6)  ilog^S. 

33.  What  is  the  value  of  \  log39-2  log^S  +  log^a? 

34.  If  a  number  is  not  an  exact  power  of  10,  what  kind  of 
number  is  its  common  logarithm  ? 

36.  Define  characteristic ;  mantissa. 

36.  How  do  you  obtain  the  characteristics  of  the  common 
logarithms  of  numbers? 

37.  Give  the  characteristics  of  the  following  logarithms  : 
log  628.75;  log  1.864;  log  .00031 ;  log  .681;  log  6931.7. 


410  ALGEBRA 

38.  Does  a  change  in  the  position  of  the  decimal  point  in 
a  number  affect  the  value  of  the  mantissa  of  its  logarithm  ? 
What  determines  the  value  of  the  mantissa  ?    Why  is  this  ? 

39.  Explain  how  to  find  the  mantissa  by  using  the  table  of 
this  book,  when  the  given  number  has  (a)  less  than  three 
digits;  (b)  three  digits;  (c)  more  than  three  digits.  Upon 
what  principle  does  this  last  depend  ?  Is  the  result  absolutely 
correct  ? 

40.  Find  the  logarithms  of  61 ;  372 ;  4  ;  3180  ;  96.4  ;  132.67  ; 
4166.8;  1.726;  .065;  .0002;  .68532. 

41.  Explain  how  to  find  the  number  corresponding  to  a 
given  logarithm,  {a)  when  the  given  mantissa  is  given  in  the 
table,  (b)  when  the  given  mantissa  is  not  given  in  the  table. 

42.  Find  the  numbers  whose  logarithms  are  2.3385  ;  1.8998  ; 
3.7528  ;  3.8594  ;  .6300  ;  T.4835. 

43.  Define  cologarithms.  When  would  you  use  the  colog- 
arithm  of  a  number?    Are  cologarithms  necessary  ? 

44.  Find  colog  6.73. 

45.  What  is  the  advantage  in  using  logarithms  in  long 
computations  ? 

"46.  By  use  of  logarithms  find  32.61  X7.26--403. 

47.  By  use  of  logarithms  find  i^671.4. 

48.  By  use  of  logarithms  solve 

(«)  3*"— 13-3^  +  36  =  0.  {b)  30^+145x^52^-23.^ 

49.  Transform  |/  ^-^  into  a  form  adapted  to  computation 
by  tables  whea  a,  h,  and  c  are  definite  numbers. 


/ 


TABLE  OF  MANTISSAS 


411 


N. 

0 

I 

2 

3 

•4 

5 

6 

7 

8 

9 

10 

0000 

0043 

0086 

0138 

0170 

0212 

0253 

0294 

03^ 

0374 

ii 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0793 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1073 

1106 

13 

1189 

1173 

1206 

1339 

1371 

1303 

1335 

1367 

1399 

1430 

1  14 

1461 

1493 

1523 

1553 

1584 

1614 

1644L 

1673 

1703 

1733 

1  15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

3014 

i  16. 

2041 

2068 

2095 

3133 

8148 

2175 

2201 

2227 

3353 

3379 

17 

2304 

3330 

235o 

3380 

2405 

2430 

2455 

2480 

2504 

8539 

18 

2553 

2577 

2601 

3635 

3648 

2672 

2695 

2718 

2T4^ 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

2iU 

.3010 

3033 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3223 

3>43 

3363 

3384 

3304 

3324 

3345 

3365 

3385 

3404. 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3830 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4300 

4216 

4233 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4363 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4503 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4943 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5093 

5105 

5119 

5132 

5145 

5159 

5172 

iT 

5185 

5198 

5311 

5334 

5237 

5350 

5263 

5276 

5289 

5302 

34 

5315 

5338 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

-sa- 

5798 

5809 
5933 

5831 
5933 

5833 
5944 

5843 
5955 

5855 
5966 

5866 
5977 

5877 
5988 

5888 
5999 

5899 
6010 

39 

5911 

40 

6021 

6031 

6043 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6333 

42 

6233 

6343 

6353 

6363 

6274 

6284 

6294 

6304 

0314 

6335 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6533 

45 

6532 

6543 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6713 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6808 

48 

6812 

6831 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6930 

6938 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093. 

7101 

7110 

7118 

7136 

7135 

7143 

7153 

52 

7160 

7168^ 

7ir7 

7185 

7193 

7202 

7310 

7218 

7226 

7335 

53 

7243 

7351 

7359 

7267 

7275 

7284 

7393 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7373 

7380 

7388 

7396 

'ii\ 

41^ 

ALGEBRA 

N. 
55 

0 

1 

2 

3 

4- 

5 

6 

7 

8 

9 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

E8 

7G;J4 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

77G0 

7767 

7774 

69 

7783 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

785a 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

79^3 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

STJ^ 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

81^6 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

855.';' 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

87^2 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9C25 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

959a 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

'92 

9643 

9647 

9652 

9657 

9661 

9066 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

995C 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

«'n 


APPENDIX 


A.     SQUARE  AND  CUBE  ROOTS  BY  FORMULA. 
(SUPPLEMENTARY  TO  CHAPTER  VII). 

1.  In  Chapter  VII  the  processes  of  extracting  the  square 
and  cube  roots  of  expressions  by  inspection  were  discussed. 
We  shall  here  show  how  to  extract  the  square  root  or  the 
cube  root  of  a  polynomial  by  use  of  a  formula.  This  is  a  gen- 
eral process  which  may  be  used  in  case  the  method  by  inspec- 
tion fails.  We  may  then  show  how  to  extract  the  square  root 
or  cube  root  of  any  arithmetical  number  by  use  of  a  formula. 

2.  Square  roots  of  polynomials.  The  square  root  of  any  pol- 
ynoniial  will  he  a  polyno7mal. 

If  the  root  has  only  two  terms,  it  will  be  of  the  form  a-Vb. 
In  that  case  the  rule  for  obtaining  the  root  comes  from  the 
identity 

The  work  is  usually  arranged  as  follows  : 

a2  +  2aZ»  +  d'  |a  +  ^,  root 

j^ 

la^h  I    2ab-^h- 

2ab-\-b'=  2ab  +  b'' 


When  arranged  in  descending  powers  of  a,  the  first  term  of  the 
root,  a,  is  the  square  root  of  the  first  term  of  the  polynomial,  a^. 
Subtracting  a"^  from  the  polynomial,  we  get  the  remainder 
2ab+b'^.     Evidently  the  second  term  of  the  root,  &,  may  be  ob- 

1 


2  APPENDIX 

tained  by  dividing  the  first  term  of  this  remainder,  2a&,  by  2a, 
OT  twice  the  part  of  the  root  already  found.  The  divisor  2a  is 
called  the  trial  divisor.  If  to  the  trial  divisor  we  add  &,  the  new 
term  of  the  root,  we  get  2a  +  b  ;  and  if  this  in  turn  be  multiplied 
by  6,  the  product,  2ab  +  b^,  will  be  the  above  remainder.  Hence, 
2a +  6  is  the  whole  or  complete  divisor.  And  a  +  ?>  is  the  entire 
square  root. 

The  above  process  will  determine  the  square  root  of  any 
polynomial  whose  square  root  is  a  binomial. 

Example  1.     Find  the  square  root  of  9a?^— 12xi/  +  4i/2. 

Here,  as  in  all  cases,  the  given  polynomial  should  first  be  ar- 
ranged in  descending  or  ascending  powers  of  some  letter.  This 
expression  is  already  written  in  descending  powers  of  x.  The 
work  will  be  as  follows : 


a^  +  2ab  +  b^=  9x'-12xy-\-4y^ 

a'=  9x' 


a  +  b 
3x—2y,  the  root 


2a  +  b=6x-2y 
2ab  +  b'= 


-12xy  +  4y' 
—  12xy  +  4y^ 


Since  9x^—12xy  +  4y^  is  of  the/or?7i  a"^  +  2ab  +  b\  therefore  a^=9x\ 
Hence,  a=Sx,  the  first  term  of  the  root.  Subtracting  9x^  we 
have  for  remainder,  —12xy  +  Ay'^. 

Now  2a,  the  trial  divisor,  becomes  6ic.  Dividing  —12xy,  the 
first  term  of  the  remainder,  by  6x,  we  obtain  —2y,  the  value  of 
b  of  the  formula.  Adding  this  to  6x,  the  complete  divisor  be- 
comes 6a;— 2^/.  Multiplying  this  by  —2y,  we  have  —12xy  +  4:yK 
Subtracting  this  from  the  remainder  —12xy  +  4:y^  leaves  no 
remainder. 

Hence,  the  entire  root  is  3x—2y. 

If  the  root  contains  three  terms,  we  have  by  grouping  terms, 
the  identity 


APPENDIX  3 

Hence,  when  the  root  has  three  terms^  the  first  two  may  he 
found  as  above  ;  then  hy  grouping  terms  these  two  tnay  he  used 
as  one,  and  the  third  term  ohtained  hy  a  repetition  of  the  p>ro- 
cess  used  to  ohtain  the  second  term,. 

In  like  manner,  when  a  root  contains  four  terms  the  first  three 
may  he  used  as  one,  and  the  fourth  ohtained  by  another  repetition 
of  the  process  used  to  ohtain  the  second.  This  process  may  he 
extetided  to  any  number  of  terms. 

Note. — The  student  should  take  care  first  properly  to  arrange  the 
order  of  the  terms  in  any  case,  either  in  ascending  or  descending  powers 
of  some  letter.  Each  remainder  should  also  he  arranged  like  the  original 
expression. 

Example  2.     Extract  the  square  root  of  16x*  + 1  — 3a?+24a^  +  Sa?^. 

First  arrange  in  descending  powers  of  x.  The  work  is  as 
follows  : 

a    +b    +c 

4:X^  +  SX  —  ^ 


a'= 

16x*  +  2^x^  +  5x'- 
16x' 

-3a?  +  i 

2a  +  b=8x^  +  Sx 
2ab  +  ¥= 

24x'  +  5a?2- 

-3x  +  i 

2{a  +  b)+c=Sx^  +  6x- 
2cla  +  b)  +  c^= 

-\ 

-40^=^- 
-4x2- 

-3x  +  i 
-3x  +  i 

Since  a^=\^x^,  the  first  term  is  4x^ 

Hence,  the  trial  divisor,  2a,  is  8x^. 

Dividing  24ar*  by  this  gives  3x,  the  second*  term  of  the  root. 

Adding  Sx  to  the  trial  divisor  gives  the  complete  divisor, 
8x^  +  3x. 

Multiplying  the  complete  divisor  by  Sx  gives  24:X^  +  9x^. 

Subtracting  this  product  from  the  first  remainder  leaves 
— 4a?^— 3a?4-5,  the  second  remainder. 

Now  using  the  root  found,  4^x^  +  3x  as  one  term,  the  new  trial 
divisor  becomes  8x^  +  Qx. 

Dividing  —  4x^  first  term  of  the  remainder,  by  8x^,  the  first 
term  of  the  trial  divisor,  gives  —  |,  the  third  term  of  the  root. 


4,  APPENDIX 

Then  the  complete  divisor  becomes  Sx^  +  Qx—^. 

Multiplying  this  by  —  ^,  and  subtracting,  the  third  remainder 
becomes  zero. 

Hence,  the  entire  root  is  4a?^  +  3j?— ^. 

Note. — Since  every  expression  has  two  square  roots,  differing  only 
i!i  sign,  a  second  expression  for  the  root  in  any  case  may  be  obtained 
by  changing  tlie  signs  of  the  terms  in  the  root  found.  Thus,  in  Ex- 
ample 2,  another  root  is  ^—dx—ix^. 

EXERCISE  1. 

Extract  the  square  roots  of  the  following : 

1.  Ax'  +  4x  +  l.  5.  8«-4a^  +  a*  +  4. 

2.  ic^  +  25y-  — lOicy  6.  l  +  %x—x''  +  Sx'  —  2x'  +  x\ 

3.  ^x'  +  x'-2x'  +  4:-4x.  7.  49aj«  +  42£c«-19£c*- 12.^^  +  4. 

4.  4a*  +  49-3a'-70a  +  20«l        8.  x*-2x'i/-2xf  +  SxY  +  y\ 

9.  x'-Qx^>/  +  UxY-12xi/'  +  4:i/\ 

10.  y*  +  4y  +  4y^  +  2y  +  4  +  i,.      12.  x^-^Sx  +  12-^ +  i, 
11    o  .  «'  I  o  A  I  A4  ,  2^'  I  ^'        19      2  lla^^3«^^9a* 

Find  to  three  terms  the  approximate  square  roots  of : 
14.  1  +  x.  15.  1-2CC.  16.  a'  +  l.  17.  m^  +  3. 

3.  Square  roots  of  arithmetical  numbers.  A^ii/  arithmetical 
number  is  in  nature  a  polynomial. 

Thus,  5263=5000  +  200  +  60  +  3 

.=  5-1000  +  2-100  +  6-10  +  3 
=  5-103  +  2-102  +  6-10  +  3. 


APPENDIX  5 

Hence,  the  square  root  of  an  arithmetical  number  will  be  ob- 
tained in  practically  the  same  manner  as  the  square  root  of  a 
polynomial. 

The  squares  of  the  numbers  1,  2,  3,  •  •  *  *  9,  10,  are  1,  4,  9,  •  •  •  • 
81,  100,  respectively.  Hence,  the  square  root  of  an  integer  of 
one  or  tivo  figures  is  a  number  of  one  figure. 

The  squares  of  the  number  10,  11,  •  •  •  •  99,  100,  are  100, 
121,  •  •  •  •  9801,  10000,  respectively.  Hence,  the  square  root  of 
an  integer  of  three  or /our  figures  is  a  number  of  two  figures. 

Likewise,  the  square  root  of  an  integer  otfive  or  six  figures  is  a 
number  of  three  figures.     And  so  on. 

Therefore,  ^/*  the  figures  of  an  integer  are  marked  off  from 
right  to  left  in  groups  of  two^  the  number  of  figures  in  the  square 
root  will  he  equal  to  the  number  of  groups^  any  one  figure 
remaining  on  the  left  being  counted  as  a  group. 

Thus,  marked  off  into  groups,  21904  becomes  2'19'04'.  Since 
there  are  three  groups,  the  square  root  of  21904  will  contain  three 
figures. 

If  the  square  root  of  a  number  be  a  number  of  two  figures, 
we  may  denote  the  tens  of  the  root  by  a  and  the  ones  by  b. 
Then  the  root  \fill  be  represented  by  a-^b.  Hence,  the  given 
number  will  be  represented  by  a^-^^ab  +  b"^. 

The  use  of  the  identity 


\/a'  +  'lab^b^  =  a-\-b 

to  extract  square  roots  of  arithmetical  numbers  is  best  shown 
by  examples. 

Example  1.     Find  the  square  root  of  3969. 

Pointing  off,  we  have  39'69.  Since  there  are  two  groups^  the 
root  must  be  a  number  of  two  figures.  Since  60^  is  less  than 
3969,  and  70^  is  greater  than  3969,  the  root  must  lie  between  60 
and  70;  i.e.,  the  tens'  figure  of  the  root  is  6,  the  square  root  of 


Q  APPENDIX 

the  greatest  square  in  the  left-hand  group  of  the  given  number. 
Therefore,  in  the  above  identity,  a  denotes  6  tens,  or  60. 
The  work  may  then  be  performed  as  follows  : 

I  a  +  b 
39'69'  |60  +  3=63,  root 
a''=  36  00 


2a=120 
2a +  6= 123 
2ab+b'= 


369 
369 


Since  a=60,  a^=SeOO. 
Subtracting  3600,  the  remainder  is  369. 
The  trial  divisor^  2a,  becomes  120. 

Dividing  369  by  120  gives  approximately  3.     Hence,  &is  proba- 
bly 3. 
Therefore,  the  complete  divisor,  2a  4- 6,  becomes  123. 
Multiplying  123  by  3  gives  369, 

Subtracting  this  from  the  first  remainder  leaves  zero. 
Hence,  60  +  3,  or  63,  is  the  required  root. 

If  the  square  root  be  a  number  of  three  figures,  we  have,  by 
grouping  terms, 

(a  +  b  +  cy={a  +  by  +  2(a^b)c  +  c\ 

Hence,  when  the  root  is  a  member  of  three  figures.,  the  first  two 
may  he  found  as  above ;  then  their  sum  may  he  used  as  one, 
and  the  third  term  obtained  by  a  repetition  of  the  process  used  to 
obtain  the  second. 

In  like  maimer.,  when  a  root  contains  four  figures.,  the  sum  of 
the  first  three  may  be  used  as  one,  and  the  fourth  obtained  hy 
another  repetition  of  the  process  used  to  obtain  the  second.  The 
process  may  be  extended  to  any  number  of  figures. 


APPENDIX  7 

Example  2.     Find  the  square  root  of  203401. 

Pointing  off  we  have  20'34'01'.     Hence,  there  are  three  figures 
in  the  root.     The  work  is  as  follows : 


20'34'01' 
a"^       16  00  00 


a-\-h-\-c 
400  +  50  +  1=451,  root 


2a  =  800 
2a  +  6=850 
2ah  +  h''= 


43401 
42500 


2(a  +  6)=900 
2(a  +  6)  +  c=901 
2c{a  +  h)  +  c''= 


901 
901 


The  largest  square  in  20  is  16.     Hence,  a^= 160000.     Therefore, 
a  =  400. 

Subtracting  160000  leaves  43401  for  remainder. 

The  first  tiHal  divisor,  2a,  is  800. 

Dividing  43401  by  800  gives  approximately  50. 

Hence,  the  complete  divisor,  2a +  b,  is  850.  * 

Multiplying  this  by  50,  we  get  42500. 
.     Subtracting  this  from  43401  leaves  901  for  second  remainder. 

Now  adding  400  and  50  gives  450,  a +  6. 

The  second  trial  divisor,  2{a  +  b),  is  900. 

Dividing  901  by  900  gives  approximately  1,  the  third  figure  of 
the  root. 

Hence,  the  complete  divisor,  2{a  +  b)  +  c,  is  901. 

Multiplying  this  by  1,  we  get  901. 

Subtracting  this  from  the  second  remainder  leaves  0. 

Hence,  the  required  root  is  451. 

When  the  square  root  of  a  number  has  decimal  places^  the 
number  itself  will  have  tioice  as  many. 

Thus,  (.23)2=. 0529. 


8 


APPENDIX 


Therefore,  to  mark  off  a  number  which  contains  a  decimal^ 
begin  at  the  decimal  point  and  mark  to  the  left  and  to  the  rights 
putting  two  figures  in  each  group. 

Thus,  723.618  will  become  7'23'.61'80'. 

Example  3.     Find  the  square  root  of  50.9796. 

a+     6+    c 
7.00 +  .10 +  .04=7. 14,  root 


50'.97'96 
a^=              49.00  00 

2a= 14. 00 
2a  +  6=:14.10 
2ah  +  h''= 

1.97  96 
1.4100 

2(a  +  6)  =  14.20 
2(a  +  6)  +  c=14.24 
2c(a  +  6)  +  c'= 

.5696 
.5696 

An  approximation  to  the  value  of  the  square  root  of  a  number 
which  is  not  a  perfect  square  may  be  obtained  to  any  degree  of 
accuracy  desired. 

Note. — For  convenience  in  writing,  each  new  trial  divisor,  which  is 
the  sum  of  the  parts  or  the  root  already  found,  may  be  denoted,  respec- 
tively, by  a,  a',  a",  etc.,  and  each  new  term  denoted  by  h,  h',  h",  etc., 
as  in  the  following  example. 

Example  4.     Find  to  three  decimal  places  the  square  root  of  2. 
Since  we  want  three  decimal  places  it  is  convenient  to  annex 
6  ciphers.     The  work  is  as  follows: 


2.'00'00'00' 
1.00  00  00 


1  +  .4  +  . 01  +  . 004=1. 414 


2a=2. 
2a  +  6=2.4 
2ah^h''= 

1.00 
.96 

2a'=2.8 
2a'+6'=2.81 
2a'h'  +  h'^= 

.04 
.0281 

2a"=2.S2 
2a"  +  &"=2.824 
2a"h"  +  b"'= 

.0119 
.011296 

APPENDIX 


EXERCISE  II. 


Find  the  square  root  o^ : 

I.  7396.         2.  1849.         3.  26244.         4.  41209.         5.  12.25. 
6.  146.41.      7.  125.44.      8.  4.6225.        9.  .026244.    10.  2611.21. 

Find  to  three  decimal  places  the  square  root  of : 

II.  5.     12.  3.     13.  .6.     14.  105.     15.  371.     16.  .75.     17.  11.8. 

4.  Cube  roots  of  polynomials.  If  the  cube  root  of  a  polynomial 
has  only  tico  terms,  it  will  take  the  form,  a-\-b,  and  the  rule  for 
extracting  the  cube  root  will  come  from,  the  identity  __ 


The  work  is  usually  arranged  as  follows  : 

d^  +  ?,ci^h-{-?>ab'^  +  Jf    \a  +  h,  root 
a^  ■       "■ 

3a2  +  3a6  +  62 


^a^h  +  ?>ah^-^lf: 


'Sa'b  +  3ab'  +  ¥ 
'Sa'b  +  Sab'  +  h^ 


When  arranged  in  descending  powers  of  a,  the  first  term  of  the 
root,  a,  is  the  cube  root  of  the  ^rs^  term  of  the  polynomial,  a'\ 

Subtracting  a^  from  the  polynomial  leaves  the  remainder 
Sa''b  +  3ab''  +  b\ 

The  second  term  of  the  root,  6,  may  be  obtained  by  dividing 
the  first  term  of  this  remainder,  3a^6,  by  3a^,  or  three  times  the 
square  of  the  part  of  the  root  already  found. 

This  divisor,  3a^  is  the  trial  divisor. 

If  to  the  trial  divisor  we  add  3a6,  or  three  times  the  product  of 
the  new  term  of  the  root  by  the  old  part  of  the  root,  and  &^  or  the 
square  of  the  neiv  term  of  the  root^  we  obtain  3a^  +  3a6  +  b^  ;  and 
if  this  in  turn  be  multiplied  by  6,  the  product,  Sa^b  +  Sab^  +  b^, 
will  be  the  above  remainder. 


10  APPENDIX 

Hence,  Sa^-\-^ab+b^  is  the  complete  divisor. 

If  the  complete  divisor  be  multiplied  by  b  and  the  product 
subtracted  from  the  first  remainder,  the  second  remainder  becomes 
zero. 

The  above  process  will  determine  the  cube  root  of  any 
polynomial  whose  cube  root  is  a  binomial. 

Example  1.     Find  the  cube  root  of  8x^—S6x*y^  +  54x^y*—27y^. 
This  is  arranged  in  descending  powers  of  x.     The  work  is  as 
follows : 

I    a  +  b 
83C^-S6xY  +  UxY-27y^    \2x'-Sy\  root 
a»=  8x^ 


Sa'=12x' 
Sa'  +  Sab  +  b'=12x'-18xY  +  ^y' 
3aH7+3ab'-\-b'= 


-36xY  +  54^V— 272/^ 
-36;rY  +  54a;V-27?/« 


Since  a'=8a?*,  then  a=2x^  the  first  term  of  the  root. 

Subtracting  8a^  leaves  —36xY  +  54x^y*—27y^. 

The  trial  divisor,  3a^,  becomes  12x*. 

Dividing  —SQx^y^,  the  first  term  of  the  remainder,  by  12a:* 
gives  —  32/^  the  second  term  of  the  root. 

3a6,  three  times  the  product  of  2x^  and  —  3?/^  is  —18x^y\  The 
b^  of  the  formula,  the  square  of  the  new  term  —Sy^,  is  9?/*. 

Hence,  the  complete  divisor  becomes  12x*—18xY  +  9y^. 

Multiplying  this  by  —3y^  gives  —  36icV  +  54xV— 272/« ;  and 
subtracting  this  from  the  first  remainder  leaves  0. 

Hence,  the  cube  root  is  2x^—3y^. 

When  the  root  has  three  or  more  terms,  by  grouping  the 
terms  of  the  root  already  found,  these  terms  may  be  used  as  one, 
and  the  next  term  found  by  a  repetition  of  the  process  used  to 
obtain  the  second. 


APPENDIX 
Example  2.     Find  the  cube  root  of 


11 


I      a+b+c 

x^-6x^+l5x*-20af  +  15x^-6x-\-l     \x'-2x+l 


a'=        x^ 

-63(^  +  15x*-20x'  +  15x'- 
-6x^  +  12x*-  8x^ 

-Qx  +  1 

a'=a  +  b=x^—2x 

Sa'''=3x*-12.x^  +  12x^ 
3a''-\-t^a'c  +  c^=3x*—12x^  +  15x'—6x+l 

3^*-12ie  +  15x2- 
3x'-12x^  +  15x'- 

-6x+l 
-607+1 

EXERCISE  III. 


Find  the  cube  root  of 


1.  x'  +  Qx'i/  +  12x}/-i-Sy^. 

2.  x'-Sx'  +  Sx'-l. 

3.  x'  +  9xy  +  27xy^27i/', 

4.  27a'-10Sa'b'  +  U4:Ctb*-Q4b\ 

5.  l-6a  +  12a^— 8a\ 

6.  x'-Sx'  +  Qx'—7x'  +  Qx^-Sx-^l. 

7.  l~9a  +  SSa'-QSa'  +  Q6a'-S6a'  +  Sa\ 

8.  S  +  x^^9x'  +  Sx'  +  Ux^  +  12x  +  lW. 

9.  a'  +  Ua  + -,-112  + ——  -12a\ 

10.  4ic=^-9ic^-6iB-l+ic«-6£c^  +  9a;*. 

11.  a;«  +  3aj*  -  54a; +  28ic'-9£c'- 6x^-27. 

a;'    ^y  ,^y''   y^ 
^^'  27""6"  •  "T'S"- 


19  APPENDIX 

5.  Cube  roots  of  arithmetical  numbers. 

The  cubes  of  the  numbers  1,  2,  3  •  •  •  •  •  9,  10,  are  1,  8,  27 
729,  1000,  respectively.  Hence,  the  cube  root  of  an  in- 
teger of  one,  two  or  three  figures  is  a  number  of  one  figure. 

The  cubes  of  the  numbers  10,   11 99,   100,  are  1000, 

1331 970290,    1000000,   respectively.     Hence,    the    cube 

root  of  an  integer  of  four^  five  or  six  figures  is  a  number  of  two 
figures. 

Likewise,  the  cube  root  of  an  integer  of  seven,  eight  or  nine 
figures  is  a  number  of  three  figures.     And  so  on. 

Therefore,  if  the  figures  of  an  integer  are  marked  off  from 
right  to  left  in  groups  of  three^  the  number  of  figures  in  the  cube 
root  icill  be  equal  to  the  number  of  groups^  one  or  two  figures 
remaining  o?i  the  left  being  counted  as  a  group. 

Thus,  marked  off  into  groups,  2515456  becomes  2'515'456'. 
Since  there  are  three  groups,  the  cube  root  of  2515456  will  contain 
three  figures,  i.e.,  ones,  tens  and  hundreds. 

If  the  cube  root  of  a  number  be  a  number  of  two  figures,  we 
may  denote  the  tens  of  the  root  by  a  and  the  o?ies  by  ^,  and 
hence,  represent  the  root  by  a  +  b.  Hence,  the  given  number 
is  represented  by  a^  +  3a^b  +  ^ab''  +  b\ 

Therefore,  the  cube  root  of  the  number  may  be  obtained  by 
use  of  the  identity 


^a'  +  Sa'b  +  ^ab'  +  b'  =  a  +  b. 

The  process  is  best  shown  by  examples. 

Example  1.     Find  the  cube  root  of  39304. 

Pointing  off,  we  have  39'304'.  Hence,  the  root  will  be  a  num- 
ber of  two  figures.  Since  30^  is  less  than  39304,  and  40^  is  greater 
than  39304,  the  root  lies  between  30  and  40;  i.  e.,  the  tens'  figure 
of  the  root  is  3,  the  cube  root  of  the  greatest  cube  in  the  left-hand 
group  of  the  given  number.  Therefore,  in  the  above  identity  a 
denotes  3  tens,  or  30. 


APPENDIX 

The  work  is  then  performed  as  follows: 

39'304' 
a'=  27  000 


13 


30  +  4=34, 


root 


3a2=2700 
Sa' +  3ab  +  b' =307  Q 
Sa''b  +  3ab^  +  b^= 


12304 


12304 


Since  a=30,  a^=27000. 
Subtracting  27000  leaves  12304. 
The  trial  divisor^  3a^  becomes  2700. 

Dividing  12304  by  2700  gives  approximately  4.  Hence,  b  is 
probably  4. 

The  complete  divisor,  3a^  +  3a6  +  6^,  becomes  3076. 
Multiplying  3076  by  4  gives  12304. 
Subtracting  this  from  the  first  remainder  leaves  0. 
Hence,  30  +  4,  or  34,  is  the  required  root. 

When  the  root  is  a  72  umber  of  three  figures  the  first  tv^o  may 
he  found  as  above  ;  then  their  sum  may  be  used  as  one  number^ 
and  the  third  obtained  by  a  repetition  of  the  process  used  to 
obtain  the  second.     And  so  on. 

Example  2.     Find  the  cube  root  of  2515456. 

Denoting  the  parts  of  the  root  already  found  by  a,  and  a' 
respectively  and  the  new  parts  by  b  and  b\  we  have  the  following 
work  : 


3a' =30000 
3a2  +  3a&  +  ?>-^=39900 
3d'b  +  ?>ab''-\-¥= 


2'515'456' 
1  000  000 
11515456 

1197000 


!  a  +  ?)  +c 

100  +  30  +  6  =  136,  root 


3a  ■-'=50700 
3a'2  +  3a'6'  +  6''''= 53076 
3a'26'  +  3a'6'='  +  6'2= 


318456 
318456 


There  will  be  three  figures  in  the  root. 


14:  APPENDIX 

The  first  number  is  found  to  be  100. 

Then  the  first  trial  divisor  is  30000. 

Dividing  1515456  by  30000  gives  approximately  50. 

But  it  will  be  found  that  the  resulting  complete  divisor^  when 
multiplied  by  50,  will  give  a  number  greater  than  1515456. 
Hence,  50  is  too  large.  It  will  be  seen  by  trial  also  that  40  is  too 
large.     Therefore  we  use  30. 

Then  the  complete  divisor  becomes  39900. 

Multiplying  39900  by  30  gives  1197000. 

Subtracting  gives  a  second  remainder  318456. 

Now  letting  a'  be  130,  the  trial  divisor,  3a'^  becomes  50700. 

Dividing  318456  by  50700  gives  approximately  6. 

The  second  complete  divisor,  3a'^  +  3a'6'  +  6'^,  becomes  53076. 

Multiplying  53076  by  6  gives  318456. 

Subtracting  this  from  first  remainder  leaves  0. 

Hence  the  cube  root  is  136. 

For  reasons  similar  to  those  given  in  §  3,  to  mark  of  a 
miniber  which  contains  a  decimal^  begin  at  the  decimal  point 
and  mark  to  the  left  and  to  the  rights  putting  three  figures  in  each 
group.  If  necessary,  ciphers  may  be  annexed  to  the  right  of 
the  decimal  point. 

An  approximate  value  of  the  cube  root  of  a  number  which  is 
not  a  perfect  cube  may  be  obtained  to  any  required  degree  of 
accuracy. 

Example  3.     Find  the  cube  root  of  1860.867. 

1'860'.867'    [10. +  2. +  .3:^12.3,  root 
a^=  1 000 .  : 


3a^=300. 
3a2  +  3a6  +  6'=364. 
3a26  +  3a62  4-6'= 


860.867 

728. 


3a'^=432. 
3a'2  +  3a'6'  +  6''=442.89 


132.867 
132.867 


APPENDIX 


15 


EXERCISE  IV. 


Find  the  cube  root  of  : 

I.  103823.  2.  2744.  3.  15625. 

5.  571787.  6.  340068392.        7.  523606616. 

9.  328.509.  10.  41.063625. 

Find  to  two  decimal  places  the  cube  root  of  : 

II.  2.  12.  10.  13.  106.  14.  3.7. 


4.  148877. 
8.  59.319. 


15.  .15 


16 


APPENDIX 


B.    HIGHEST    COMMON  FACTORS  AND   LOWEST  COM- 
MON MULTIPLES  BY  DIVISION. 

(SUPPLEMENTARY  TO  CHAPTER  X). 

1.  In  Chapter  X  we  showed  how,  by  factoring  the  expres- 
sions, to  find  the  highest  common  factor^  {11.  C.  F.)  or  the 
lowest  common  multiple  (X.  C.  M.),  of  two  or  more  expressions. 

The  highest  common  factor  of  two  expressions  may  be  ob- 
tained also  by  a  division  process,  known  as  the  Euclidian  Pro- 
cess, which  we  shall  here  discuss.  It  may  be  used  in  those 
cases  where  the  factors  are  not  readily  found.    • 

2.  H.  C.  F.  by  division.  The  process  involves  the  following 
principle  : 

If  A  =  BQ^  R,  ichere  A,  B,  Q,  and  B  are  integral  expres- 
sions in  the  same  general  number,  then  the  H.  C.  F.  of  A  and  B 
is  the  same  as  the  IL  C.  F.  of  B  and  R. 

This  principle  is  established  by  showing  that  every  common 
factor  of  A  and  5  is  a  common  factor  of  B  and  i2,  and  every 
common  factor  of  B  and  J2  is  a  common  factor  of  A  and  B. 

Since  A=J5^+i2,  it  follows  from  the  Distributive  Law  that 
every  common  factor  of  B  and  jR  is  a  factor  of  A,  and  hence  is  a 
common  factor  of  A  and  B. 

Again,  R—A—BQ,  and  therefore  it  follows  as  above  that  every 
common  factor  of  A  and  jB  is  a  factor  of  R,  and  hence  is  a  com- 
mon factor  of  B  and  R.     This  proves  the  principle. 

Now  suppose  that  A  and  B  are  two  integral  expressions  whose 
highest  common  factor  is  required,  the  degree  of  JB  being  not 
higher  than  that  of  A. 


APPENDIX  17 

Divide  A  by  B,  and  call  the  quotient  ^i,  and  the  remainder  R^. 
Divide  B  by  R^,  and  call  the  quotient  Q2,  and  the  remainder  R^. 
Divide  Ri  by  R2,  and  call  the  quotient  ^3,  and  tlie  remainder 
R^;  and  so  on. 

NoAV  since  the  remainder  must  be  of  lower  degree  in  the 
letter  of  arrangement  than  the  divisor,  by  this  process  the 
successive  remainders  must  diminish  in  degree.  Hence,  finally 
a  point  will  be  reached  when  either  the  remainder  is  0,  or 
does  not  contain  the  letter  of  arrangement. 

Jf  the  remainder  is  0,  then  the  last  climsor  is  the  required 
highest  common  factor. 

For,  since  the  dividend  equals  the  product  of  the  divisor  and 
quotient  plus  the  remainder,  we  have  from  the  above  divisions 

J3=R,Q,  +  R,, 


i?„_2  —  i?n-l  Qri  +  ^n-> 


where  i?,^  is  the  last  remainder. 

From  these  equations  it  follows  by  the  preceding  principle 
that  the  H,  C,  F.  of  B  and  B^  is  the  same  as  the  11.  G.  F.  of  A 
and  i?;  the  H.  C.  F.  of  B^  and  B^  is  the  H.  C.  F.  of  B  and 
i?„  and  hence  of  A  and  B ;  the  II.  C.  F.  of  B.,  and  B^  is  the 
II  C.  F.  of  7?i  and  B^,  and  hence  of  A  and  B ;  and  so  on. 
That  is,  the  II  C.  F.  of  A  and  B  is  the  II  C.  F.  of  any  two 
consecutive  remainders  in  this  succession  of  divisions. 

But  if  A\=0,  the  H.  C.  F.  of  B,,_^  and  i?„  is  B,,_,  itself. 
Therefore,  the  II.  C.  F.  of  A  and  B  is  B„_i  the  last  divisor. 

If  the  last  remainder  is  not  0,  but  merely  free  of  the  letter  of 


;|^g  APPENDIX 

arrangement,  then   there  is  no    common  factor  containing  the 
letter  of  arrangement. 

For,  if  7?„  is  merely  free  of  the  letter  of  arrangement,  then 
J?„_i  and  i?„  can  have  no  common  factor  which  contains  the 
letter  of  arrangement.  Therefore,  A  and  B  can  have  no  com- 
mon factor  which  contains  the  letter  of  arrangement. 

*3.  It  is  clear  that  the  above  process  consists  simply  of  re- 
placing the  two  given  polynomials  by  two  new  polynomials 
which  have  the  same  H.  C.  F.,  then  replacing  these  by  two 
other  polynomials  which  have'  the  same  IT.  C.  .F.,  and  so  on. 
Hence,  it  is  allowable  at  any  stage  of  the  work  to  multiply  either 
the  dividend  or  divisor  by  any  expression  u^hich  has  not  a  factor 
belonging  to  the  other.  Likewise,  it  is  allovxible  to  remove  from 
either  the  dividend  or  divisor  a  factor  10 J lich  is  not  common  to 
both.  A  factor  common  to  both  dividend  and  divisor  may  be 
rernoved,  provided  that  it  is  introduced  into  the  H.  C.  F.  as  finally 
found. 

Example  1.     Find  the  H.  C.  F.  of  Gx^  +  llir  +  S  and  2x'  +  7x  +  Q. 

6a^2  +  lla;+   3    |2a?^  +  7a?+6       2x^  +  70.' +  6     |2jg  +  3,  H.  C.  F. 
6x"'^  +  21x-4-18       3  2x'  +  ^x  x  +  2 


—  5     )-10.r-15  4x  +  6 

2x+   3  4a^  +  6 


In  the  first  division  we  obtain  the  remainder  —10a?— 15.  From 
this  remainder  we  remove  the  factor  —5,  since  —5  is  not  a  factor 
of  the  divisor  2x'^  +  7x+Q. 

Now  the  H.  C.  F.  of  the  given  expressions  is  the  H.  C  F.  of 
2ic'  +  7x  +  6  and  2x  +  3.  Dividing  2x'^  +  7x-\-Q  by  2a? +  3  leaves  no 
remainder.     Hence,  their  H.  C.  F.  is  2a? +  3  itself. 

Example  2.  Find  the  H.  C.  F.  of  2x'  +  3(^-7x'-20x  and 
^af-12x^  +  5x. 


APPENDIX  19 

Removing  the  common  factor  x  the  work  is  as  follows : 


2ie  +  a?2-7a?-20 
2 

40?^- 

o? 

|4a?=^- 

7 

-12a?+5 
-12a?  +  5 

4a?2-12a?  +  5 
4x=^-10a? 

—  2a?  +  5 

—  2x+5 

|2a?-5 
|2x-l 

4ar*+   2a-''-14a?-40 
4:.x^-12x'+   6x 

14ar^— 19a?— 40 
2 

28a?2-38a?-80 
28x*2-84a?  +  35 
23)     46a?-115 
2a?-     5 

H.  C.  F=x{2x-5),  ov2x^-5x. 

Here,  in  order  to  avoid  fractions,  we  multiply  2x^  +  x'^—7x—2Q 
by  2  before  dividing. 

We  then  seek  the  H.  C.  F.  of  4a?2— 12a?  +  5  and  the  remainder 
14a?=^-19a?-40. 

Again,  to  avoid  fractions,  we  multiply  14a?^— 19a?— 40  by  2. 

The  second  division  gives  the  remainder  46a?— 115.  From  this 
we  remove  the  factor  23,  since  23  is  not  a  factor  of  4a?^— 12a?+5. 

We  then  seek  the  H.  C.  F.  of  20-- 5  and  4a;=^  — 12a?+  5.  Dividing 
leaves  no  remainder.     Hence,  their  H.  C.  F.  is  2a?— 5  itself. 

Now  to  get  the  H.  C.  F.  of  the  original  expressions  we  must 
multiply  2a;— 5  by  the  factor  x  which  was  removed  in  the 
beginning. 

Example  3.  Find  the  H.  C.  F.  of  ^oc'y-^xy'  +  y^  and 
3a?^ — Sx^y  +  xy"^ — y^. 

Since  y  is  a  factor  of  the  first  expression,  but  not  of  the  second, 
it  may  be  removed. 


20 


APPENDIX 


We     now     seek      the     H.     C.     F.      of     4:X^—5xy+y^    and 
3ar^ — 3x^y  +  xy^ — y^.     The  work  is  as  follows : 


Sa^ — Sx^y + xy^ — y^ 

___^ 4 

12a?' — 1 2x^y  +  4xy^ — 4?/^ 
12a^-15x'y  +  Sxy^ 

3x^y-\-xy^—4:y'^ 

4 

12a?^2/+   4£C2/^— 16?/^ 

19^/'')      19xy'-19y' 
x-y 

4xl—5xy  +  y^ 
4iX^—4txy 


4:X^—5xy  +  y^ 


3x 


4:X^—5xy  +  y^ 


32/ 


-y,  H.  C.  F. 


4x-y 


—  xy  +  y' 

—  xy  +  y"^ 

4.  The  H.  C.  P.  of  three  or  more  expressions.  The  11.  C.  F.  of 
three  or  more  expressions  may  be  obtained  by  finding  the 
H.  C.  F.  of  any  two  of  the  expressions,  then  finding  the  11.  C.  F. 
of  this  IT.  C.  F.  and  the  third  expression ;  and  so  on. 

Example  1.     Find  the  H.    C.   F.  of  a'  +  a'-Ua-24:,  a^-Sa' 
—  6a  +  8,  anda^  +  4aHa— 6. 
Thei?.  C.F.old'  +  a'-Ua-24:Siuda^-3a'-Qa  +  8isa''-2a-8. 


The  H.  C.  F.  of  a'-2a-8  and  d'  +  4a'  +  a- 
Hence,  a  +  2  is  the  H.  C.  F.  required. 


■6  is  a  +  2. 


EXEBCISE 

FindtheH.  C.  F.  of: 

1.  a^-Sx-2,  x^-x^-4:. 

2.  aj'^  +  ic-e,  ar'-2a;^-£c  +  2. 

3.  ar''  +  7ic^  +  5a;-l,  Zx^ -^  hx' +  x-1. 

4.  2a2-5a  +  2,  4a^  +  12«2-a-3. 

5.  4aj^-12a;-27,  eaj'^-Slaj  +  lS. 


APPENDIX  21 

6.  2a^  +  9a'-6a-6,  Sa'  +  10a'-2Sa  +  10. 

7.  Sx'  +  x-2,  4x'  +  2x'-x  +  l. 

8.  2a;'-7a'-2  +  7a,  ba'  +  a'-Za'-A.a  +  A. 

9.  2m''  — 3m'  — 8?72  — 3,  3m*  — 7m^  — Sm'- m— 6. 

10.  9a=^^-22a5=^-85*,  3«='  +  13a'^>  +  12a^»l 

11.  2x''-^x''y-2xy\  2x^  +  lxhj-\-Zxy\ 

12.  a^-^alf-2h\  2a'-^a'b-ab'  +  Qb\ 

13.  a;y-6ajy +  6ecy-3a;y +  2.^/,  6a!V-15xV  +  21a;V-12a;y 

14.  2lab- 17 ab'-bab'-\-ab\  baly'-SAab'-7ab. 

15.  ic*  +  4£c^  +  4:x\  x'y  +  5a;\y  +  ^x-y. 

16.  a;^  +  3a;'  +  4,  £c='  +  2£c'-4a;  +  8. 

17.  ic='  +  3a;  +  2,  jc*-6a;=^-8aj-3. 


18. 

x'-x^^\,x'^x^-^\. 

19. 

x''-x'-%x^^\2,  x''^4x'—Zx^- 

20. 

a?-\,  2a'-a-l,  Sa'-a-2. 

21. 

2m' +  3m -5,  3m'- m- 2,  2m' 

)n  —  S. 

22.  x'  +  x-Q,x'  +  Sx'-Qx-^,  x'-2x'-x  +  2. 

23.  3«'  +  5«'  +  a— 1,  a'  +  3a  — 1— 3«^  «'  — l  +  7a'  +  5a. 

24.  a'-9a-10,  a'-30-7«,  lO  +  a'-lla. 

26.  2y'  +  Qy'  +  4y\  3?/=^  +  9y' +  9y  +  6,  3^=*  +  8y' +  5y  +  2. 

5.  L.  C.  M.  by  use  of  H.  C  F. 

The  L.  C.  M.  of  tico  exjyressions  may  be  obtained  by  cUviduiy 
their  product  by  their  H.  C.  F. 

This  may  be  proved  as  follows : 

Let  A  and   B  be   the   expressions   whose   lowest   common 
midtiple  is  required,  and  let  ^be  their  highest  common  factor. 

Then  A  and  B  may  be  written 

A  =  Hq,, 

and  B=IIq^^ 

in  which  q^  and  q^  have  no  common  factor.     (Why  ?) 


22  APPENDIX 

Hence,  AJB = Hq^Hq^ = IT{Hq^q.^. 

Now  the  L.  C.  M.  must  contain  all  factors  common  to   A 
and  i?,  all  other  factors  of  A^  and  all  other  factors  of  B. 
Hence,  the  L.  C.  M.  equals  Hq^q^. 
Therefore,  from  the  above  equation, 

AB==H(Hq,q,), 

we  see  that  the  product  of  A  and  B  equals  the  product  of 
their  11.  C.  F.  and  their  L.  C.  M  Therefore,  their  L.  C.  M. 
ma  (J  he  obtained  hy  dimding  their  product  by  their  H.  C.  F. 

6.  L.  C.  M.  of  tliree  or  more  expressions.  The  Z  CM.  of 
three  or  more  expressions  may  be  obtained  by  first  finding  the 
Z.C.M.  of  any  two  of  the  expressions,  then  finding  the 
If.  CM  of  that  result  and  a  third  expression  ;  and  so  on. 


EXERCISE  VI. 

Find  the  Z.  C  M  of : 

1.  2a'-5a4-3,  2«=^— 7a  +  5. 

2.  ^a'  +  Ua+lO,  Qa'  +  ba-U. 

3.  a^-x'-Ux  +  2i,  x'-2x'-bx-}-e. 

4.  x'-12x+lQ,  x'-4x'-x'  +  20x—20. 

5.  12a^  +  13a^  +  6a  +  l,  lQa'  +  lQa'  +  7a^l, 

6.  4:rv'-12n'  +  bn,  2n*  +  n^-7n'—207i. 

7.  x'-Sx+2,  x*-Qx'  +  Sx-Z. 

8.  2x'  +  6x-^,  2x'-{-x'-6x  +  2. 

9.  2a'  +  Sa-b,Sa'-a-2,2a'-Va-S. 

10.  a^+x-Q,x^-2x^-x  +  2,x'  +  Sx'-Qx-S. 

11.  2x'  +  Sx-2,  2£c''  +  15a;-8,  iB^+10i«  +  16. 

12.  2a*+W  +  4a\  3«^+9a^+9a+6,  Sa'  +  Sa'  +  ba+2. 


APPENDIX 


C.    FUNDAMENTAL  PROPERTIES   OF  NUMBERS. 

1.  In  the  following  sections  we  shall  establish  some  fun- 
damental laws  of  numbers  which  are  at  the  foundation  of 
arithmetic  as  well  as  of  algebra.  These  laAvs  were  assumed  in 
Chapter  II.  They  apply  to  any  numbers.  We  shall  now 
establish  these  laws  for  commensurable  mimhers.  By  the  aid 
of  the  theory  of  limits  they  can  be  shown  to  hold  for  incom- 
rnensurahle  numbers  as  well. 

We  shall  first  establish  the  laws  for  integers. 

2.  Law  of  order  in  addition.     Numbers  integers. 

•  Numbers  to  be  added  may  be  arranged  in  any  order ;  that  is, 

The  proof  of  this  law  is  based  upon  the  meaning  of  an  integer. 

{a)  When  the  numbers  are  positive  integers.  By  definition, 
positive  integers  are  merely  arithmetical  numbers,  each  repre- 
senting the  number  of  units  in  a  given  magnitude. 

Let  us  consider  two  magnitudes  of  the  same  kind,  one  con- 
taining a  units  and  the  other  b  units.  Then,  a-\^b  means  the 
number  of  units  in  the  new  magnitude  formed  by  putting  the 
second  magnitude  with  the  first.  And  b-\-a  means  the  num- 
ber of  units  in  the  new  magnitude  formed  by  putting  the 
first  magnitude  with  the  second.  But  evidently  the  total 
number  of  units  in  the  two  magnitudes  is  the  same  whether 
the  second  be  put  with  the  first  or  the  first  put  with  the 
second.     Hence,  a+ ^=5  +  «. 

By  similar  reasoning,  this  law  can  be  shown  to  be  true  for 
any  number  of  positive  integers. 


24  APPENDIX 

(b)  When  the  numbers  are  negative  mtegers.  It  has  been 
proved  that  the  sum  of  any  number  of  negative  integers  is 
obtained  by  finding  the  sum  of  their  absolute  values,  as  in  (a) 
above,  and  attaching  to  this  sum  the  negative  sign.  But  by  {a) 
the  sum  is  the  same  for  any  arrangement  of  their  absolute 
values.  Hence,  the  law  is  true  also  for  any  number  of  7iegative 
integers. 

(c)  Wheii  the  numbers  are  integers^  some  i^ositive  and  some 
negative  It  has  been  proved  that  the  sum  of  two  numbers, 
one  positive  and  the  other  negative,  is  obtained  by  finding  the 
difference  of  their  absolute  values  and  attaching  to  this  the 
sign  of  the  number  having  the  greater  absolute  value.  This 
difference  would  be  the  same,  whatever  the  arrangement  of  the 
numbers.  Hence,  the  law  of  order  in  addition  liolds  for  two 
integers^  one  positive  and  the  other  negative. 

By  (a),  (^),  and  (c),  it  follows  that  an  integer,  positive  or 
negative,  may  be  brought  to  any  position  without  changing  the 
value  of  the  sum.  Hence,  the  law  is  true  for  any  number  of 
integers^  some  positive  and  some  negative. 

Note.— This  law  is  also  called  the  commutative  law  of  addition. 

3.  Law  of  grouping  in  addition.    Numbers  integers. 

Numbers  to  be  added  may  be  grouped  in  any  manner  ;  that  is, 

a-h6  +  c  =  a-h(6  +  c). 
Since  a,  5,  c,  are  integers,  we  have  by  §  2, 

a-\-b-^c=b  +  c  +  a 
=  (J  +  c)  +  a 

Similar  reasoning  will  apply  in  the  case  of  any  number  of 
integers. 

Note.— This  law  is  also  called  the  associative  law  of  addition. 


APPENDIX  25 

4.  Law  of  order  in  multiplication.    Numbers  integers. 

The  product  of  two  or  more  numhers  is  not  changed  by  chang- 
ing the  order  in  ichich  the  midtiplications  are  j)erfornied ;  that  is, 

abc  =  acb  =  bac,  etc. 

(«)  When  «,  ^,  c,  are  positive  integers,  place  a  objects  in  a 
group,  and  form  h  rows  of  c  groups  to  the  row. 

c  columns 


h  rows 


a  a  a  a 
a  a  a  a 
a     a     a     a 


Since  there  are  a  objects  in  each  group  and  b  groups  in 
each  column,  there  must  be  ab  objects  in  each  column.  And 
since  there  are  c  columns,  the  total  number  of  objects  must  be 
abc. 

In  like  manner,  there  are  ac  objects  in  each  row,  and  b 
rows.     Hence,  the  total  number  of  objects  must  be  acb. 

Hence,  abc  and  acb  represent  the  same  number  of  objects. 
Therefore,  abc=acb. 

If  now  a=l,  this  becomes   bc=cb. 

(b)  If  there  be  any  number  oi  positive  factors,  by  (a)  any 
two  consecutive  factors  may  be  interchanged  (considering  the 
product  of  all  factors  preceding  these  two  as  the  lirst  factor 
a  in  (a)  above) ;  and  by  repetition  of  the  process  of  intercliang- 
ing  two  consecutive  factors,  all  of  the  factors  may  be  arranged 
in  any  order  without  changing  the  value  of  the  product. 

(c)  The  law  hokls  if  some  of  the  factors  are  negative.  For, 
the  absolute  value  of  the  product  is  the  same  whatever  the 


26  APPENDIX 

signs  of  the  factors,  and  the  sign  of  the  product  has  been 
shown  to  depend  only  upon  the  number  of  negative  factors. 
Any  change  in  the  order  of  the  factors  could  not,  therefore, 
change  the  sign  or  absolute  value  of  the  product. 

Note.— This  law  is  also  called  the  commutative  law  of  multiplica- 
tion. 

5.  Law  of  grouping  in  multiplication.    Numbers  integers. 
Factors  may  he  grouped  in  any  manner  /  that  is, 

ahc=a{hc). 
When  «,  J,  c,  are  integers,  we  have  by  §  4, 
abc=hca 

=  {bc)a. 

=  a{bc). 

Similar  reasoning  will  apply  in  the  case  of  any  number  of 
factors. 

Note.— This  law  is  also  called  the  associative  law  of  multiplica- 
tion. 

6.  Law  of  distribution.    Numbers  integers. 

The  product  of  tic o  expressions  equals  the  sum  of  the  products 
obtained  by  multiplying  each  term  of  either  exj^ression  by  the 
other  /  that  is, 

(a  +  b  +  c)x—ax  +  6  jr  +  ex. 
(a)    When  x  is  positive. 

Since  ic=l  + 1  +  1+ to  £c  terms, 

then  (a  +  b  +  c)x={a  +  b  +  c)^{a^b  +  c)  +  {a  +  b  +  c)-\- 

to  X  terms  (Def.  of  multiplication.) 

=<^4-a  +  «+   •  •  •  •  to  a;  terms 
-]rb-\-b  +  b+     ••••tocc  terms 
+  c  +  c  +  c-{-     '  '  ■  '  to  ic  terms  §2 

=  ax  +  bx  +  cx. 


APPENDIX  27 

Similar  reasoning  will  apply  to  any  number  of  terms  in  the 
multiplicand. 

(b)    When  x  is  7tegatwe. 

Since  — £c=  — 1  — 1  — 1—  ....  to  ic  terms, 

therefore      (a  +  ^  +  c)  ( — a?)  =  —  (a  +  ^  +  c)  —  (a  +  ^»  +  c) 

—  (a  +  6  +  c)  •  •  •  •  to  a;  terms  (Def.  of  multiplication). 

=  —a  —  a—a—  •  •  •  to  £c  terms 
—h—b—b—  •  •  •  to  £c  terms 
—  c—c—c—    •  •  •  to  £c  terms    §  2. 

=  a{—x)^-b{—x)^-c{—x). 

Similar  reasoning  will  apply  to  any  number  of  terms  in  the 
multiplicand. 

7.  In  order  to  prove  the  preceding  laws  when  fractions  are 
involved,  it  is  necessary  first  to  establish  the  following  prin- 
ciples. 

m 

(1)  a—=am-^n; 

that  is,  to  multiply  any  expression  a  by  the  fraction  —  multiply 
a  by  m,  then  divide  the  product  by  n. 

This  follows  immediately  from  the  definition  of  multipli- 
cation. 

(2)  To  prove  abc  =  acb 

lohen  a  is  a  fraction,  and  b  and  c  are  integers. 

If  b  and  c  are  positive,  form  b  rows  of  the  fractions  a  with  c 
fractions  in  each  row.     Then  there  are  c  columns. 


28 


b  rows     ^ 


APPENDIX 

c 

columns 

A 

a 

a 

a     

a 

a 

a     

a 

a 

a     

Now,  by  definition  of  multiplication,  ab  represents  the  sum 
of  the  rt's  in  one  column.  And  since  there  are  c  columns,  abc 
represent  the  sum  of  all  of  the  «'s. 

Again,  ac  represents  the  sum  of  the  «'s  in  one  row.  And 
since  there  are  b  rows,  acb  represents  the  sum  of  all  of  the  a's. 

Therefore,  abc  =  acb. 

If  b  or  c,  or  both  b  and  c,  are  negative,  the  proof  folloAvs  as 
in  (c),  §  4. 


(3)     To  prove 


a  m_m  a 
b  71      n    b 


This  is  established  as  follows : 


a  ni 
~b  n 


By  (1). 


=  (y-) 


"     I, 


m-~-nnb 
—nnb 

(quotient  X  divisor  =  dividend.) 


a 
^-^■bm 


.,     m  a      - 

Also— T-no^ 

n   b 


(m  a\ 


m  a 


n   b 


bn 


By  (2). 
(quotient  X  divisor  =  dividend.) 


By  (2)  and  §  4. 


APPENDIX 

29 

= — a-^hbn 

n 

By(i) 

=  (^.«)^6.6.« 

m 

(quotient  X 

divisor  =  dividend.) 

=  m,a 

(quotient  X 

divisor  =  dividend.) 

=  €1-^1. 

§4. 

_,            a  tn      ^     m  a      . 
Hence,  -r — 710= — -rno. 
'   0  n            n   0 

Axiom  7. 

Dividing  by  5,  then  by  w, 

a  m 
Tn 

m  a 
-n'~b' 

If  n=l,  this  becomes 

a 

a 
=  m~^. 

8.  Fundamental  laws  when  fractions  are  involved. 

(1)  The  Imo  of  order  in  multiplication.  In  (3)  §  7  we  have 
established  the  law  of  order  in  multiplication  for  two  factors, 
where  one  or  both  are  fractions.  By  the  same  method  the 
law  can  be  shown  to  hold  for  any  number  of  factors. 

(2)  The  law  of  grouping  in  multiplication.,  ^fhen.  some  of  the 
numbers  are  fractions,  now  easily  follows. 

Thus,  abc=bca  By  (1). 

=  {bc)a 

=a{bc). 
The  same  reasoning  would  apply  to  any  number  of  factors. 

(3)  The  laic  of  distributioii.  We  are  now  able  to  complete  the 
proof  of  the  law  of  distribution. 


30  APPENDIX 

We  have  x(a+b+c)—aia-\-xb-{-xc,  whatever  the  expressions 
represented  by  x,  a,  b,  c,  because  the  multiplier  a  +  b  +  c  is 
obtained  by  first  taking  unity  to  form  a,  then  to  form  b,  then 
to  form  c,  and  adding  the  three  results ;  and  hence  to  obtain 
the  product  x  must  be  used  in  the  same  manner. 

Now,  since  x(a  +  b  +  c)=xa  +  xb  +  xc, 

(a  +  b  +  c)x=ax-{^bx  +  cx,  by  the  laAV  of  order. 

The  same  reasoning  would  apply  to  any  number  of  terms. 

(4)  The  law  of  order  in  addition  noio  follows.  It  has  been 
shown  that  integers  or  rational  fractions  can  be  reduced  to 
equivalent  fractions  having  a  common  denominator,  and  such 
that  the  resulting  numerators  and  denominators  are  all 
integers. 

Suppose  that  when  reduced  the  fractions  are 


etc. 


X 


Then,  f+|+f+  ....  ="  +  ^  +  f  •  •  ■  •  §50,  Chapt.  VI. 

But  the  value  oia  +  b  +  c  •  •  •  •  is  not  changed  by  changing 
the  order  of  the  terms.     Hence  the  value  of -  +  ^+-+ 

Jb  Jb  tJu 

is  not  changed  by  changing  the  order  of  its  terms. 

(5)  The  laid  of  grouping  in  addition,  when  fractions  are  in- 
volved, can  now  be  established  by  the  same  reasoning  that  was 
used  in  §  3. 


INDEX 


[Numbers  refer  to  pages.] 


Abscissa,  193. 
Absolute  term,  243. 
Addition,  1,  35,  46,  47,  146. 

of  negative  numbers,  35. 

of  fractions,  146. 

elimination  by,  183. 
Algebraic  expressions,  5. 

numbers,  32. 

sum,  35. 
Antecedent,  298. 
Arithmetical  numbers,  33. 

means,  359. 

progressions,  354,  355. 
Arrangement  of  expressions, 
Axioms,  21. 


60. 


Cologarithms,  401. 
Combinations,  342. 
Commensurable  numbers,  299. 
Common  difference,  355. 
Completing  the  square,  235. 
Complex  fractions,  155. 
Complex  numbers,  223. 
Conditional  equations,  20. 
Conjugate  surds  and  imaginaries, 

219,  226. 
Consequent,  298. 
Constants,  310. 
Continuation,  symbol  of,  56. 
Coordinates,  192. 
Continued  proportion,  304. 


Base  of  power,  12. 
Binomial,  15. 

square  of,  77. 
Binomial  theorem,  81. 

extraction  of  roots  by,  353. 

general  term  of,  349. 

proof  of,  346. 
Brace,  bracket,  etc.,  6. 

Character  of  roots,  341. 
Checks,  48. 

Clearing  of  fractions,  165. 
Coefficients,  13. 

numerical,  12. 

undetermined,  375. 


31 


Degree,  of  terms,  131. 
of  an  equation,  161. 
Difference,  1. 
Discriminant,  241. 
Distributive  law,  18. 
Division,  4,  64,  66,  153. 
Divisor,  dividend,  4. 

Elimination,  181,  182,  184,  186. 
Equations,  19. 

equivalent,  179. 

exponential,  388,  406. 

fractional,  165. 

graphic  representation,  192,  373. 

graphic  solution,  197,375,378,279. 


32 


INDEX 


inconsistent,  179. 

independent,  179. 

indeterminate,  178. 

ill  quadratic  forms,  253. 

integral,  163. 

irrational  and  radical,  161. 

linear,  164. 

quadratic,  230. 

simultaneous,  179. 

solutions  of,  178. 

symmetrical,  267. 

systems  of,  180. 
Equivalent  equations,  179. 
Evolution,  84. 
Exponents,  12. 

laws  of,  56,  64,  75,  324. 
Expressions,  5. 

literal,  13. 
Extremes  and  means,  300. 
Evaluation  of  an  expression,  6. 

Factoring,  103. 

equations  solved  by,  233. 
Factorial  -n,  340. 
Factors,  12. 

Formula,  the,  168,  169.^ 
Formula  for 

solving  quadratics,  238. 
Fourth  proportional,  300. 
Fractional  equations,  165. 

exponents,  324. 
Fractions,  71. 

complex,  155. 

partial,  383. 

Geometrical  progression,  354,  362. 

means,  365. 
Graphic,  representation  of  and 

solution  of  equations,  192,  197, 
273,  275,  278,  279. 


representation  of  imaginary 
numbers,  228. 

Harmon ical  progression,  354,  370. 
Highest  common  factor,  131. 
Homogeneous  equations,  264. 
expressions,  131. 

Identical  equations,  identities, 

20. 
Imaginary  numbers,  223. 
Incommensurable  numbers,  299. 
Inconsistent  equations,  179. 
Independent  equations,  179. 
Indeterminate  equations,  178. 

fractions,  321,  322. 
Inequalities,  291-295. 
Inserting  signs  of  grouping,  53. 
Integral  equations,  161. 

expressions,  130. 
Involution,  75. 
Irrational  numbers,  86,  212. 

Known  and  unknown  numbers, 
24. 

Law  of  order,  16. 

of  exponents,  56,  64. 

of  grouping,  16. 

of  signs,  41. 

expressed  by  an  equation,  214. 
Letter  of  arrangement,  60. 
Like  and  unlike  terms,  15. 
Linear  equations,  163. 
Literal  numbers,  9,  11. 

advantage  of,  9,  11. 
Logarithms,  388. 

characteristic  of,  395. 

computations  by,  403. 

common,  393. 


INDEX 


33 


Napierian,  393. 
Lowest  common  multiple,  135. 

Mantissa,  395,  397. 
Mean  i)ioportional,  304. 
Means,  arithmetical,  359. 

geometrical,  365. 

liarmonical,  370. 
Means  and  extremes,  300. 
Minuend,  1. 
Monomials,  15. 
Multiples,  L.  C.  M.  135. 
Multiplicand,  multiplier,  3. 
Multiplication,  3,  39,  57,  58,  150. 

Negative  exponents,  326. 

numbers,  30-38. 
Numbers,  definite,  9. 

algebraic,  33. 

commensurable,  299. 

constants  and  variables,  310. 

finite  and  infinite,  319,  320. 

general,  8,  9. 

imaginary  and  complex,  223. 

known  and  unknown,  24. 

natural,  9. 

opposite,  33. 

rational  and  irrational,  163,  212. 

real,  223. 

Opposite  numbers,  33. 
Ordinate,  192. 

Parentheses,  6. 
Partial  fractions,  383. 
Permutations,  337. 
Polynomials,  15. 

square  of,  79, 
Powers,  12. 
Prime  factors,  104. 


Progressions,  354. 

arithmetical,  354,  355. 

geometrical,  354,  362. 

liarmonical,  354,  370. 
Problems,   directions  for  solving. 

173. 
Products,  4. 

special,  92. 
Proportion,  principles,  299. 

Quadratic  equations,  230. 

graphs  of,  273,  275,  277. 

principles,  242. 

systems  of,  261. 
Quality,  signs  of,  32. 
Quotients,  5. 

special,  96. 

Radicals,  83,  86,  212. 
Ratio,  298. 

Rational  expressions,  6,  86. 
Rationalizing  factor,  220. 
Real  numbers,  223. 
Remainder  theorem,  123. 
Removal  of  si^ns  of  grouping,  51. 
Roots,  83. 

of  a  polynomial,  90. 

of  an  equation,  21. 

character  of  roots,  241. 

sum  and  product  of,  242. 

Series,  254,  378. 

convergent,  354. 

divergent,  354. 

finite,  354. 

fractions  expanded  into,  378. 

oscillating,  354. 

reversion  of,  382. 
Signs  of  grouping,  6. 
Simultaneous  equations,  179. 


34 


INDEX 


Solution  of  equations,  21,  178. 

graphic  method,  275. 

by  special  devices,  270. 
Square  of  binomial,  77. 

of  polynomial,  79. 
Square  roots,  79. 
Subtraction,  2,  37,  49. 
Subtrahend,  2. 
Surds,  86,  212. 
Symbol  of  continuation,  56. 
Symmetrical  equations,  267. 
Synthetic  division,  127. 
Systems  of  equations,  180. 

consistent  or  determinate,  180. 

defective,  269. 

equivalent,  180. 

impossible,  180,  190. 

indeterminate,  190. 

solution  of,  180. 


Theorem,  trinomial,  81. 

of  undetermined  coefficients, 
375. 

remainder,  123. 
Third  proportional,  304. 
Transposition  ,165. 
Trinomial,  15. 

Undetermined  coefficient^,  S!7b. 

theorem  of,  376.  j 

Unknown  numbers,  24. 

Variables,  310. 

limit  of,  318. 
Variation,  310. 

direct,  311. 

indirect,  311. 

joint,  312. 
Vinculum.  6. 


Terms,  14. 

Theorem,  binomial,  81,  346. 


Zero  exponent,  64. 
operations  with,  321. 


ANSWERS 

Exercise  1. 

8. 

8.                 11.  33. 

14.   i.                 17. 

44. 

20.  13. 

^. 

15.                12.   0. 

15.   38.               18. 

16. 

21.  24. 

10. 

2^                 13.  48. 

16.  4.                19. 

6. 

22.  0. 

23.  216. 

24. 

1.                  25.   18. 
Exercise  2. 

26. 

25. 

1. 

5.              4.   27. 

7. 

230.          10..  ^V- 

13. 

23. 

16.    2,    r 

2. 

11.             5.  40. 

8. 

37.            11.  28. 

14. 

h 

17.  17.. 

3. 

270.          6.  C6. 

9. 

24.            12.  49. 

15. 

liV 

18.  576. 

19.  20tli;  X  taken  as  a  factor  20  times. 

20.  4;  5;  1.  21.  7;  x\  3. 


Exercise  4. 

5.  35a6a?2^. 

12.  14a'. 

19.  ll^-'^a?. 

6.  30a&c. 

13,  ^\x^y. 

20.  484. 

7.  2xyz. 

14.  'S2x2j^. 

21.  693. 

8.  14abcxy. 

15.  mah.    , 

22.  24.8. 

9.  15a^bx^y. 

16.  28irV- 

23.  x;  x^y;  y\ 

10.  iabcxyz. 

17.  lOax. 

24.  x2;  a;  6. 

11.  15a. 

18.  loab^ 

Exercise  5. 

25.  7?i2;  3m;  21. 

1.  3.        3.  2. 

5. 

3. 

7.  2.           9.  2. 

11. 

2.         13.  I.         15.  1. 

2.  7.        4.  4. 

6. 

3. 

8.   1.         10.  1. 
Exercise  6. 

12. 

3.        14.  5.        16.  2. 

1.  8  boys. 

3.  10  yrs.  and  30  yrs. 

5.  16,  48. 

2.  $3. 

4.  45,  80. 

6.  8,  15. 

2  ANSWERS  [Ex.  6-9 

7.  B,  $250.     A,  $500.     C,  $800.  8.  4  cows;  12  hogs. 

9.  112  yds.  by  224  yds.  10.  8  hrs.  H-  H  ^^^'  Per  hr. 


12.  64. 

16. 

65,72 

20.  18  mi. 

24.  12,  24. 

13.  9. 

17. 

32,48 

21.  12  hrs. 

25.  $24,000. 

14.  48  lbs. 

18. 

57,58 

,  59.          22.  If  hrs. 

26.  7. 

15.  24,  39. 

,      19. 

6  da. 

23.  15,  20,  : 

25. 

27.  5  yrs.,  35  yrs. 

28. 

12  mi. 

29.  3  lbs. 

30. 

A,  25;  B,  27. 

Exercise  7. 

1.  3. 

4.  -27. 

7.   -3V               10. 

-9. 

13.  14. 

2.  7. 

5.  2i. 

8.  -.24.-^         11. 

-2. 

14.  4. 

3.  -22. 

6.  0. 

9.  9.                    12. 

—  5. 

15.  14. 

16.  -3. 

18. 

6. 

19.-39. 

23. 

-200  lbs;  -50  lbs. 

24.  25  ft ; 

+5; 

+25; 

-5;  +15.          25.  $210;  - 

-$160. 

Exercise  8. 

1.  -14. 

7. 

540. 

13. 

81.            19.  24. 

25. 

-3.          31.  f. 

2.  -14. 

8. 

-280 

14. 

-64.        20.  108. 

26. 

-xV        32.  -54.-^ 

3.  14. 

9. 

ri 

15. 

-1.          21.  576. 

27. 

-2.5.       33.   -5i. 

4.  14. 

10. 

1. 

16. 

216.          22.   -162. 

28. 

-12.        34.  8. 

5.  30. 

11. 

0. 

17. 

.432.          23.  -9.- 

.29. 

-i.          35.  -192. 

6.  -12. 

12. 

32. 

18. 

72.            24.  9. 
Exercise  9. 

30. 

18.            36.  -28f. 

1.  3a?. 

3. 

-4c3. 

5.  IQax^. 

7.  10F<^. 

2.  -4a26. 

4. 

-3a5cd.              6.  lA. 

8.  -B^. 

9.  2lpq. 

13. 

-23m27i2. 

17. 

—  5ahc, 

10.  86f.4C.  14.  -X.  18.  ^yz. 

11.  -5ofiy\  15.  -18^3.  19.  12a-9x. 

12.  -Slpqr.  16.  5a^l^c.  20.  Qah-ldxy. 

21.^  -2a5c2+9a26c+7a52c. 
28.  3a2+262.  24.   {2+a+h)xyz. 

23.  (3a+56-7c)a;2,  25.  (-7c+2+Sa)y\ 


Ex.  10-13] 


ANSWERS 


1.  -dx-^y+5z. 

2.  7a— c. 

7.  2oc^+2x. 

8.  4a2. 

9.  2i)d^+6x^y—Qxy^—2y^. 


Exercise  10. 

3.  -3P+dQ+4R-5S.   5.  15p+6(/-S 

4.  dac—ixy.  6.  Sab+Qbc. 

10.  23Vi-|6^ic. 

11.  5x^+i^x-2. 

12.  -9a?4+4a?34-2£c2+7a7+4. 


9.  7x-10y. 

10.    -a75-ic3  +  .x2. 

n.  4a26+2a62+268. 
12.  2AB+5xy-'7PQ. 


Exercise  11. 

1.  6a%\  5.  136?/2. 

2.  —  CoTi/.  6.  24x^y'^z. 

3.  aic2.  7.  2x-7y+10z. 

4.  -lOabc.  8.  4a+14a6— 7c. 
13.  cc2-ll£c+13.  .  14.  x''+x^+oc^+x*+2o(^+x+l 

15.  a*4-3;rS+?-7+3c2s5. 

16.  a;3-2a;2+^-l;  a_2^24.^_l.  _2x^+x-l. 
17.  _as-a862+4a253_2ic.  18.  a.'3_^7^2^_^,^_l. 

19.  9.125a-7.6a;2-6.25?n3-5??i2-l. 
20.  —x^+x^+x—2.  24.  —x^+Qx^—Qx—4.      27.  —0734.407—6. 


22. 

2aH5-l|a:2+10 

25. 

a,'8_2a;2+8a7-2. 

28.  36. 

23. 

7a3+2a2-3a- 

-7.         2S. 

a;3-4a7+6. 

29.  -9. 

30. 

3. 

32.   -3. 

34.  16. 

36.  16. 

31. 

-18. 

33.  8. 

35.  26. 
Exercise  12. 

37.  8. 

1. 

3a- 6.  _ 

2. 

507+5?/. 

7.  -2b. 

3.  52+4^( 

4. 

4a-2b. 

5. 

3aj3-7£c2+l. 

8.  -2a?3- 

-4x^+x-4:. 

6. 

-3(K8-a;2?/+6a;2/2+22/8. 

9.  7x'+Sxy-5y^-2y». 

10. 

a-b. 

12.  5-2/. 

14.  -4a;+3a.         16.  2. 

11. 

15a-26. 

13.  e+ic. 

15.  1. 

17.  x: 

1.  x^-oc^+oc^-ix^-x+l), 

2.  ax—{by+cz—dw). 


Exercise  13. 

3.  a-(-25+3o-4d). 

4.  -(10e-5/+9r). 


ANSWERS 


[Ex.  13-16 


5.  (2+a)c(^+{b-S)x^+(6-c)x. 

6.  7+{5-2a)x+{l+Qb)x^+{d-4a)x^. 

7.  (a-'d)x*+  (2+a)c(^+  {l-b)x^+  {d-c)x-l, 

8.  _(5_2)7/-(l-a)?/+5. 

9.  -(-p+q-r)y-{dq-2p)y^-sy*. 

10.  -(6x+2)y^-{d-x^)y^-(dx^-5)y. 

11.  10+(2+b+c)x-(a+2)x'^. 

12.  (a+Q)x^-5x.  13.  5ii;2-3a7-l. 

14.  {l+a)x^+  (a.-b)x^+  (b+c+l)x+4:. 

15.  (a-3)ic-^+(a2+a-3)a'2+2aa;.        17.  (a-b)x'2+ib-c)x+c+d. 

16.  .Tio+(2-a+&)ic5+?>+c-d.  18.   (b+2)x^-{c+S)x+5-a, 

19.  pa:;3_p^2_(gjfr7')a;_^_g. 

Exercise  14. 

6.  4a^b^c^. 

7.  -^^5^4. 

8.  -P^Q^. 

12 


1.  -«7;,i2. 

2.  60a^66. 

3.  210a;V- 

5.  -3a2364c7c?5. 


y.  —pn^'n 
10.  14.21875a;62,3. 


48pV'- 

2i^22^11. 

2(a+b)5. 
-15a3(6+c)*. 


11. 


11. 
12. 
13. 

14. 

16.  x'^\ 
17.  Qa\  19.   (a3)io. 

Exercise  15. 

5 .  da^y^zw — Sxf^j/zHv  +  Soc^yzw^. 

6.  10p2g2,._15pg2^2+20p2gr2. 

7.  ^35-^53. 

8.  -80ir8+12ar5-8^. 

10.  1 5a363c3ic22/2_  Qa^b'2c^x'^y'^+da^bcx^^+ dal^cx^y^+dabc^x^y^z. 

11 .  —  5a%22/4 + a^x'^y^— -ja^xy* + ^'^x^yK 
-14?/.  16.  -6ir-12?/. 


1.  a^b^-2a^b^+a^b\ 

2.  ic'^— a^+.r^— a;4. 

3.  -6a6624.5a5j>3_2a364. 

4.  Ax'^y^+Qx^y^-Wx'Y- 

9.  -Ix^y^z^-lx^y^z^ 


15.   -26c. 


1.  a2+2a5+52. 


17.  2ic3+3ic^?/-5ir2/2. 
20.  24ic2-30iC2/. 

Exercise  16. 
2.  a2-62. 


18.  2ab^-2a^b. 

19.  7a?4-42/*. 


3.  6a;24-iC2/-22/2. 


Ex.  16J  ANSWERS  5 

5.  307*4-072-4.  7.  20p2g2-14pV-6p2?-2. 

8.  .t2-h5o;4-6.  10.  16a2-49.  12.  12x^y^ -Sc-^d^. 

9.  4a;2+4a;-35.  11.  ic2-l00.  13.  ia^_|f£c2+^. 

14.  ^3ga2_|a5-^\62.  17.  0.94a2-5.55a6-3.64&2. 

15.  ^od»-^x^y+^\xy^-j%y^.  18.  a^x^-b^if-bcyz+acxz. 

16.  5.625372+ 15.375^^-1 1.257/2.  19.  p^q^+2pq^r-p^qr+q^r^-pqr\ 

20.  ait;4-5a;-f-ca;+a?/+b?/+e2/+a-2;+62;+c2;. 
21.  2a7*+irS-8.T2+23;^7_i2.  22.  a;5+£c4+ic3_a;2-a;-l. 

23.  4a77-2a76+7£)75— 7a74+7a;3-7j;2+3o?-5.  . 

24.  x8+;;c4^_i,  27.  a4+a252+54.  W/^^^ 

25.^-a^2+<^23^_^^ ^  28.  ic^-l.  "— «^~     ' 

2a^2«®-«'^'^-14«i^*+19«I'^^-^^    23.  ^2^2_4^8C_ 4^3(7+ 16^502. 

30.  —  2p2c2+3gr2(^2_2gc?re-r2e2+pgcd+3jpcre. 

31.  -2a;8+3a;c6+2aj6-6a2a;2-3«ic2+4a£c4+2a2x4. 
•  32.  a4-4a36+6a262-_4a53+54, 

33.  £c«4-aa7^— 4a2£t^— 46t3a'3+4a4o72+3a5o7. 

34.  a262_ 52c24_26c2fZ + 2b2cd-c^d^-2bc(P- bM^. 

35.  ai4-2aio6"3+2a666-a2&9."  41.  |a3-ia2&+i.|a52- ^^^63. 

36.  a;"+i-f-a?«?/+a7y«+2/«+^-  40  a?*     llx^a^    a* 

37.  a^2«_2^2n,  ■  30       900       30' 

38.  a6<^+a*^'^62«+a2o53._,.55c,  43.  a73-a72+|£c-|. 

39.  x^"+^-x^''+^y^+x^y»-^-y^.  U.  Sx^+8x^-\^x^+%^x^-\^x+3. 

40.  a;»  +  2_£pn+l4.2it:n_^^«_l  +  ^H-2^  45.     ^9^;;p4_4  3£p2_|„^_ 

46.  2.8x'*-7.36;r3+5.7a;2+4.16a'-10.24. 

47.  (r2«— 07*2/*— ^"Z/'+Z/*^'-  ^-  ^^^4- 1007^4-35072+5007 +24. 

48.  07»+'"  +  07'«2/"  +  07"2/'"  +  2/"+'".  55.    1 07^  +  ^0727/+ ^077/2 +^',^3, 

49.  x^+2x^—x—2.  56.  07*"— 2/*".  ^ 

50.  o?8-l.  57.  a6— &6. 

51.  6a3+a25-lla62_663.  68.  4a6. 

52.  x^-1.  69.  072-907+6. 

53.  ai2_5i2,  60.  2x^-2xy-2y^. 

61.  6£c8+12a;2-14a;-4.       62.  5a'^-oa^b-2a^b^-ab»+b*+a^b'^+a*b^-ab\ 


5 

ANSWERS 

[Ex.  17-19 

• 

Exercise  17. 

1. 

aK 

6. 

-Aa^bc^.          11    -is^t. 

16.  -2st. 

2. 

a^l^. 

7. 

ip\                  12.  5a7. 

17.  9r+2. 

3. 

-2a863. 

8. 

-12ia4.           13.  a^. 

18.   ^fz. 

4. 

-^xy^. 

9. 

7a262..              14.  -V^»r 

'. 

19.  |s». 

5. 

465c. 

10. 

-|??i%p3,        15.  £c<-?/«-i. 
Exercise  18. 

20.   Vs2M. 

1. 

x-^l. 

5.  10a4+964. 

9. 

2x'^y^+ix^y—%x^y^. 

2. 

xy-l+4x\ 

6.  a2-2a&+3b2. 

10. 

2x-|2/+if^2/2. 

3. 

-x^+5-dx^. 

7.  —a+b+c. 

11. 

a— 6+c. 

4. 

-3x^+2x^ 

8.  |j;7-2^'V.  • 

12. 

a2+a6+62. 

13.  —la^—'^-ixy'^+'^-ix^y. 

15.  5a7«-2a;»+i.       le,  a«_a2«. 


14.  13£c4-20.8a?3+39. 

17.  a7«+2/".  18.  l+£f2+ic4. 


1.  3a+l.  4. 

2.  5.2^+1.  5. 

3.  y+1.  6. 

10.  s(^-\-xhj+xy'^+y^. 

11.  x^—xy+y^. 

14.  a^-a262_^54.  19. 

15.  1207-1.  20. 

16.  x+1.  21. 

17.  x^—2xy+y^.  22. 

18.  .r4+£c2+i.  23. 

29.  —d-dx—x"^. 

30.  a2_,_2a6+a+4?>2_25+l 

31.  a;3_4^2+ii£t;_24. 

32.  x—y+z. 

33.  £C3_,_^2y_|_a72/24-2/8. 

34.  a4+3a3+9a2+27a+81. 

35.  2a2+9a-5. 

36.  x^+ocf^y+x^y^+y^. 


Exercise  19. 

5a+l. 

3a2-5a-2. 

x^+2xy+y^. 


7.  c-4. 

8.  a-5. 

9.  a2+a5+62. 


12.  x^—x^y+x'^y^—xy^+y*. 

13.  a4+a252_^54. 
it'4+a73-|-.x2+a;H-l,       24.  a;*— 072+!. 
2ic4+aic2_3a2,  25.  ^y-L 
2a2-5a+7.  26.   o^— 2a:2rt+2a?a2_a3. 
£C^+£c2//2+i/4.  27.  l-3£c+2a;2-a^. 
a;2— 6^7+9.  28.  a;2+2ir2/+%^- 

37.  a2+5a+6. 

38.  x'^—xy—xz+y^+z'^—yz. 

39.  it'2+2a??/+2/2— 0^2;— ?/2;+;2;2. 

40.  ^a2--Ja6+|62. 

41.  ?i+^+a^62+4a253+i664. 
lb       4 

42.  |£C2— ^.T2/+2/2_ 

43.  ^x^-lx^y+^xy^. 


Ex.  19-21] 


ANSWERS 


44.  a2'»+2a'"6'"— 62'n. 

46.  2a;«-r3a;»-i— 5ic»-2. 

1.  a" 

2.  a28. 

3.  -aiofois. 

4.  a?6y/8. 

5.  x^hf^z-^. 

6.  a6a?3o?/i8. 

7.  Oojio. 

9.  343aV- 

10.  ^2x^^y^. 

11.  -32a20o;^. 

12.  aj2o?/i2. 


45.    a^«  — £c2a^a_j_^<i^2<i_|^3a^ 

47.  ^£c4»-iic2»^"4-i-^2«.        48.  2a+6-3c. 


Exercise  20. 

13.  -125a9a'«^9. 

14.  —  243mi0yi^ 

15.  16d8£c44. 

a8 


21. 


27ai5 
125636* 


16 


512- 


a?2o  • 

_a28^49 
2/42    • 


20    256.Tl2y4 
•  2401a2068' 


18. 
19. 


22    1000000p«0q24,.12 

23.  a2«. 

24.  ai0'»5i2m^ 

25.  £c6«^*";22n^ 

26.  — ir*"+22/i*'*+'^, 

(,^30x52 16a; 


27. 
28. 


32^«- 


1.  x^+2xy+y^.  12. 

2.  4a2+4a6+b2.  13. 

3.  4a2+12a&+9&2.  U. 

4.  Ax^+ixhf+y^.  15. 

5.  a2_2a5+b2.  I6. 

6.  x^-2xy+y'^.  17. 

7.  4ic2- 12^7/4-92/2.  18. 

8.  a*-2a262+b4.  19 

9.  4a;6-4^V+2/^-  20. 

10.  x»-10x^y^+25yK  21. 

11.  16a^-24a;2^2+92^.  22. 

34.  49a2+28a6+452. 

35.  9a%*+6a262a;2+54. 

38.  4-4xy+x^^.  40. 

39.  (r2?/*-2a73|/34-a^2/2.  41. 

44.  a;2»_2aj»2/''+2/''". 


Exercise  21. 

a;2+2ii;+l.  23.  1664-4062+25. 

4x2+"l2a;+9.  24.  x^-2x^+l. 

9ic*+24a;2+16.  25.  a^-2a^+l. 

25£t'«+10a;3+l.  26.  x^-2x''^y2+yK 

a8+20a4+100.  27.  x^^+2x^+x^. 

4a4+28a2a;+49a;2.      28.  x^y^—Uxy+id. 
64a6+48as>^2_f_9^.     29.  a^-20.r2+100. 
x^y^+iaxy+ia^.       30.  a,'8— 1007*4-25. 
in^-Qm+9.  31.  x^-2x^+x*. 

i;j4_6„j,2_|_9.  32.  4a7*+4r8H-ic2. 

m^n^—imny+iy^.     33.  25ir*— 100:^+076, 

36.  4a;2-126a?7/4-962^2. 

37.  4a262+16a62c+1662c2. 
ir2o_|_2icio+l.  42.  £c2«_^2a;»+l. 
ic24_2£pi2+l.  43.  a;2»-2a7»+l. 

45.  4a;2»+i2a;«a»+9a2'«. 


8  ANSWERS  [Ex.  21-23 

46.  a2n+6_3rt«+8|ia+8+^2<.+6.  a* _2a^a^    ofi 


47.  4a:2'»+2+4?iar2»+J+n2aj2». 


it'i'>  ,   .T^  ,    a?* 


49.    a2»62n-2_2a2n-l&2»-l4.a2«-252». 

53.  4a2+4G(b+?>2+4ac+26c+c2. 
a2    2ac    cf 

°"-   b^     bd     d^'  54.  9a2+6a6-6ac+6^-26c+ca- 

55.  16-16a-86+4a2+4a&+62. 

56.  16-246+ 9&2_24c+186c+9c2. 

67.  if-i^+c^+ab-^-^+'^. 

Exercise    22. 
1.  l4-2a;+3ic2+2a^+a;*.  2.  a8+2a6+5a*+4a2+4. 

3.  Ax^+12x^+2oa^+2ix^+iexf^. 

4.  a;W+2a;8+3^+2a^+2cc4+2a?3+a;2+2a7+l. 

5.  icc»-12x^+25x^-2^x^+16. 

6.  a2+4624-9c24-i6d24.25e2-4a&+6ac-8ac?+10ae-126c+16&d-206e- 

24c<i+30ce-40de. 
7^  1^2a;+3i»2+a74+3a^+2a7'+.T8. 

8.  25i^i2+70ir8-20aj«+49iC*-28a;2+4.    * 

9.  x^+y^+z^+yi^+2xy—2xz+2xw—2y^+2yw—2zw. 

10.  16ir*+24aa^+ci2aj2_6a3a;+a4.  ' 

11.  93(fi+4y^+25bf+a^-12c^y^+mx^b-Qx^a^-20y%+iy»a^-l(la^b. 

12.  6*+16a2c2+100-8a62c+20b2_80ac. 

13.  a^b^+b»c^+c^d'^+a^d^-2ab'^c+4abcd—2a^bd-2bc^d-2acd^. 

14.  a«+6«H-fi«+d64-2a%8_2a8c8-2a3df_253c3_253^3+2c3#. 

15.  4x^- 20a^+ 53a;2-y:9 + 9a4- 12/12^^2 _ 70^+ 30a2a; - 42a2. 

16.  l—2x+dx^—4QC^+5x^—Q^+5x^—ix^+da^-2c(^+x^K 

17.  a?2»+?/2»+2;2'»+2a7«?/»+2a;»2;'»4-22/«2:'*. 

18.  a;2»  +  ^2»+2_j.2;2n+4_2a;»y«+l_1.2£t;»2»+2_22/»+l;^*+2. 

Exercise    23. 

1.  a^_|-9a;82,_|.36a;72/2+84a^2/8+126a;52^-+-126aj4^4-84icY+36a%7^ 
9xy^+y\ 


Ex.  23-24]  ANSWERS 

2.  x*—4a^y+Qx^^—ixy^+y^. 

3.  x^-Qx^a+16a^a^-20a^a^+15x^a^—Qxa^+a^. 

4.  x^^+i0x^y+i5x^y^+120x-^y^+2i0x^y^+252x^y^+210x*y^+120a^y'^+ 

45x^y»+10xy^+y'^^. 

5.  12r)oc^+150x^y+mxy^+8yK  6.  Gix^+iSx^y^+Ux^y^+y^. 

7 .  x^-  -  Qx^^y'^ + 1 5x»y*  -  20x^y^  + 1 5a^?/8  -  6ic22/io + y^\ 

8.  256a^-768£c6?/2+864a^2/4_432aj22/64.8l2/8. 

9.  a:;84-5a^+10a;3+10£c2+5£C+l.  10.  l+^a+Qa^+ia^+a*. 

11.  £c«-6a^+15ic*-20a^+15ic2-6a7+l. 

12.  ir5-10a?4+40.T3-80£c2+80^-32. 

13.  l+7y+21?/2+352/8+35i/4+2l2/6+T^+2/^ 

14.  a^+2.T3?/+fa;2,,2+i;:c,/34.^i^^.  15.  2^7a^_^x^y^2xy^-^\y». 
16.  ^\afi+^\x^y^+lx^y^+^x^+j\y^ 

--    a8    3a2c    3ac2     c3  .q    8^    36^    54^    27^ 

18.   ^-§fL^?4^'-§2^+16.  20.  Ax^-2xy+4. 

yi      y^      y^        y  4 

21.  a^+fa^z/+|a?2?/2+ffa?^+^«^. 

23.    l-?^+^-20a8+60a4-96a5+64a6. 
64     8        4 

64ag    IggSb    5a^b2    5^353     135^2^4  _243CTb5    729bg 
729     "2^         3  2    "^      64  256    "*"4096* 

„-    x"'    boc^y    6x^y^_5x^y^  ,5xy^ y^ 

•   32     ~W^    36         54   "^  162     243* 

Exercise  24. 

1.  ±3a45.                           8.  -5w6n2.  15.  2a^. 

2.  ±4iC2/8.                          9.  4x^,  16.  -3ic8y. 

3.  ±7aa:2?/6,                     10.  fa?^.  17.  -^aft*. 

4.  ±15»i6n2.                    11.   ±2a62.  18.  ±2a;2z/8. 

"  20.   ±x^y^z. 

6.  -2a;?/^.  13.   ±5a78|/.  ^^a 

7.  -a62c8.  14.   ±|a:2/2gr8.  ^1.    ±^. 


10  ANSWERS  [Ex.  24-27 


6ctS 
96** 


ANSWERS 

24     5«' 

26. 
27. 

±11/5. 

25.    ±f. 

28. 

±il/2. 

Exercise    25. 

23.  fa'. 


16.  07+5  or  —x—5.  31.  a^c^+^Z^  or  —ax^—y^. 

17.  a;+6  or  -ir-6.  32.  Sxy-ia^b^  or  4a263_3a^^ 

18.  x+8  or  -ir-8.  33.  l-3a?  or  Sx-l. 

19.  07+9  or  -x-9.  34    ^^^  ,^  ^^  -l.T-i2/. 

20.  07-10  or  —07+10. 

21.  07—15  or  —07+15. 

22.  a+25  or  -a-25.  36.    ^_-  or  --g. 

23.  2o7+7  or  —  2o7— 7. 

24.  3o7— 5  or  —  3o7+5. 

25.  4a— 6  or  — 4a+6. 


35.  |x2-|?/2  or  |2/2-|ic2. 


37.    -+lor---l. 


4o;     7    „_    7      4o? 


26.  9o;2_^2  or  -9o;2-2.  ^®-   -j      4^  ^^  4^~T* 

27.  Ilo--lorl-llo73.  .4   .,  _«^_4 

28.  13a+76or-13a-7&.  "2"^a2            2     02* 

29.  3o:2_2a.2  or  2a.2_3^.2.  ^^^        ^^ 

30.  5a6+a;  or  -Sa^-ojl  12    x^  12'^07*' 

Exercise    26. 

1.  a;+4.  3.  2o7+3.  5.  ix'^—dy^. 

2.  07—5.  4.  3a— 4&.  '  6.  a— 5  or  6— a. 

7.  07+5  or  —07—5.  10.  2x+Sy. 

8.  3a3-468  or  4b3_3a8.  11.  3a - 62  or  b^- 3a. 

9.  x—2.  12.  a^+ 2a  or —073— 2a. 

Exercise    27. 

1.  x^-y^  7.  4a2-9.  13.  x^y^-a^jf^. 

2.  m^-n^.  8.  9o;2-42/2.  14.  x^y^-z^'^. 

3.  072-25.                           9.  25w2-16n2.  15.  4icio-25^. 
4^  a2-100.                       10.  aj*-2/*.  16.  9a*-496". 

6.  ^2_9.  11.  ^6-66.  17.  l6a2^_25&42/2. 

6.  a^-x^.  12.  mi2-ni2.  18.  ia7*-|||/4. 


Ex.  37-30] 

ANSWERS 

11 

19.  6.25a6-2.8962. 

22.  x^-2xy+y^-U. 

^.  r*+r2+l. 

20.  ll|a*a.'4-iniii/8. 

23.  a;2-a2-10a-25. 

26.  x*-l. 

21.  a;2+2a?2/+2/2-4. 

24.  4£c2-92/2+242/-16.     27.  ss+s^+l.^- 

28.  0^16-256. 

32.    ^^ 

c2 

29.  2a722/2-}-2it?2^2+ 22/2^:2 

!_a4_2^_;z4.           b2 

"da- 

30. a2«-100a;2«. 

31.  £t;2«+2_2/2a_2. 

33         1           1 

Exercise    28. 

1.  /2+7^+i2. 

8.  S2-13.S+30. 

is.  .A6- 13^8.^42. 

2.  a;2+4a;-21. 

9.  p2g2_,_22pg+120. 

16.  p2_3p_io. 

3.  62_ii5+30. 

10.  a^+14a?2+48. 

17.  m6?i*+14m%2_^48. 

4.  ir2-8a?-20. 

11.  H-8r2+15. 

18.  a;2_i8a;+72. 

5.  a2-2a-48. 

12.  x^+Sx^-^S. 

19.  30-13a+a2. 

6.  ?n2- 137/1+22. 

13.  .T4+9ic2-36. 

20.  02+2(7-35. 

7.  aj2_9^_^oo. 

14.  x^y^-dxy^-4. 

21.  ic2n+i0a;«+21. 

22    a;2«_8a;<»  +  15. 

29.  ai2- 

-2a9-9a6+10a3-200. 

23.  a2«+2_ct«+i_3o. 

30.  afiy^ 

'+2a7S2/^-a:r*2/4-2a^?/8-3aV. 

24    B* -10 AB2C+2iA^C^.                     31.  ^2_ 

2bz+h2+12z-12b+S5. 

25.  a2+2a6+52+a+?>- 

-6.                     32.  a^+2a?3-2a;2_3a;+2. 

26.  x^—2xy+y^+10x- 

■10^/+21.           33.  x^- 

Aa^x^+Sa*. 

27.   p2_,_2pg+^2_6p_| 

6g-160.             34.  a;2n_ 

-^nyn_2\y^\ 

28.  x*+2x^+ix^+'6x- 

■18.                     35.  a;2'»+24-a7n+i2/»-i-62/2''-2. 

Exercise    29. 

1.  a;+3. 

7.  5ab-i. 

13.  l+6m2n8. 

2.  £c-3. 

8.  l+7aa;. 

14.  13a?t/3-12a*. 

3.  a2+5. 

9.  l+4a:4. 

15.  5a +1562. 

4.  63-4. 

10.  t^-1. 

16.  a»+6». 

5.  2;z;+l. 

11.  8a?2+92/2. 

17.  a?»+i-a»-i. 

6.  3a-l. 

12.  lOa-lla^. 

18.  a+6+c. 

19.  a+b+x+y. 

20.  {a+b)^-{x-y)\ 

10.  m8+m2n,+mn2+w8. 


Exercise    30. 

11.  l-2/+2/2-2/8+2/*-2^. 


12 


ANSWERS 


[Ex.  30-33 


14.  a^-a^+a^-a+l.  17.  8a^-12a^h+18a%^-27b^. 

15.  a^+x'+x^+x+l.  18.  27a3-45a2Z>2+75a64- 12566. 

16.  a2+62.  19.  a^-cc^y^+x^y^—x^y^+y^. 

22.  a^+a^2/+^2/'^+^2/'+^^2/*+^2/^+2/^- 

23.  x'^—a^y+x^y^—x^y^—x^y^—x^y^+xy^—y'^. 
24.  a^2/6+a;22/4a8+a^2a6_,_a».  25.  25a-862. 

26.    64p6  +  32p5g8^_16p4g,6_,_8p3^_|.4p2gl2_f_2pgl5_^gl8^ 

27.  a^»— a;2"2/"+a7''2/*«— ^3».  31.  a^"— a2»+a»— 1. 

Exercise    31. 

4.  ar4(a^+5). 

5.  x^y^(x^+y*). 

6.  5a2(a-26). 


1.  a;(aj2— 2^+1). 

2.  ir2/(a;+3a?2!/— 5?/). 

3.  a?(a;2-3). 

10.  Sx'^{2x^+Sy^+xy). 

11.  12a*5'»(262-3a2). 

12.  4x2(ic2+l)i 

13.  2a8(4a2+2a5+62). 
18.  ^^(l+a). 

1.  5(a;-4)2. 

2.  4(a-b)2. 

3.  a(.T8+3)2. 

4.  Mxy+h)^. 

5.  5a(2a-3)2. 


1.  {x+Q){x+7). 

2.  (x+8){x-Q). 

3.  (a?-5)(a;-4). 

4.  {x-7)ix+A). 
6.  (07+8)  (a?+9). 

6.  (a;-|-10)(aT-5). 

7.  (a;-14)(a?+4). 


7.  7a;2(4£c2+?/2). 

8.  9x^(2x^-1). 

9.  a*(a2-6a6+262). 

14.  a262c2(a253  +  52c4  +  (j3c2). 

15.  6a782/(4a722/2_2+7a?32/). 

16.  9m?i(3m2+4mn+9?i2). 

17.  14a2?/2(4?/2-a2/+2a2). 

19.  a»(a7«— 1).  20.  5a''-'^y''-'^{a+2y) 


Exercise    32. 

6.  -l(3a;-l)2. 

7.  -l(a2-4)2. 

8.  h^{5a-x^y)^' 

9.  Sxy{x-ly)^. 
10.  8(6a2+56)'^. 

16.  3a(a-6)6. 

Exercise    33. 

8.  (£c-4)(a?-8). 

9.  (a+15)(a-12). 

10.  (m-16)(m+15). 

11.  (^-21)(f4-20). 

12.  (a+l)(a+i). 

13.  (c+10)(c-7). 

14.  (6+14) (6+6). 


11.  7ix+y)^ 

12.  5a(a-6)8. 

13.  2xy(x^+y)». 

14.  -5aa?(a-36)8. 

15.  2(2a-6)*. 


15.  (c+21)(c-4). 

16.  (d-ll)(d+5). 

17.  (2+r)(l+r). 

18.  (6+s)(4-s). 

19.  (3+^)  (5-2/). 

20.  (2+a)(l-a). 

21.  (2*^-2)  (22+1). 


Ex.  33-34] 


ANSWERS 


13 


30. 
49. 
50. 
51, 
52. 
53. 
54. 
55. 
56. 
57. 
58. 
59. 
60. 

1. 

2. 

3. 

4. 

5. 
16. 
17. 
18. 
23. 
23. 
24. 


{x''-2)(x^-5). 

(x^-8)ix^+2). 

ix'2+18){x'^-10). 

(x^+l){x^+2). 

(x^-S)(x^~2). 

(0^+13)  (a^-3). 

(£C«-16)(a^+14). 

(ax+ll)(ax+2). 

(a+136)(85-a). 

(xy—Sah)  (xy—2ab). 

{xy^+9ab)  {xy^-2ah), 

(xy-5c){xy+2c). 

(x^y^+7a)(x^y^+2a). 

{x^y^-M^h)  (ic3?/3-4a86). 

(am2+5c2)(am^+6c2). 

(l_2a)(l-a). 

(l+9.T)(l-3a;). 

(a+6-f-2)(a+6+5). 

{x-y-\Q){x-y+^). 

{{x+yy+W}  {{x+y)^+^}. 


31.  (b7/-10)(6?/+3).        40.  (a66-7)(a66+5). 

32.  (ab+10)(«6+20).     41.  {xy^+n){xy^-4:). 

33.  {xy-8){xy-2(i).       42.   (p5Jg4_l)(p2g4_2). 

34.  (mri+10)(mn— 6).    43.  {x+2y){x+y). 

35.  (abcH-15)(a6c-2).   44.   {x-^ij){x-2y). 

36.  (j72/2;-10)(a7^2f-9).  45.  (a7+72/)(ic+107/). 

37.  (x2j/2-l)(a;22/2_2).    46.   (a+136)(a-36). 

38.  (a^.v8+3)(aV+ll).47.   (a-206)  (a+26). 

39.  (aj42/*-14)  (07*2/4+9).  48.  (5+3a)(6-a). 

61.  {a+h—x—y){a+h-2x-2y). 

62.  2(a;-12)(iC+7). 

63.  3(£C+3)(a;-2). 

64.  5(a?+4)(a;+5). 

65.  2(a;-25)(a;+8). 

66.  a{x+1[){x—2). 

67.  a2(a;+7)(£C-5). 

68.  x{x+la){x—Qa). 

69.  307(07— 24^)  (aj+ 42/). 

70.  0722/2(0^+72/)  (07+22/). 

71.  2o7(o72/+26)(o72/+5). 

72.  2(10-'a;)(ll+a;). 


(2ir+l)(oj+2). 

(307+1)  (0^+2). 

(2o7+5)(o;+2). 

(2o;-l)(cc+3). 

(2o;-3)(o;-l). 

(7a3-l)(2a3+l). 

(3a2ic24.3)(2a2.:c2+3) 

(a6c+4)(5a&c— 1). 

3a  (207+3)  (07+1). 

22/(2oj+7)(9oj-l). 


Exercise    34. 

6.  (5oj+l)(o;-2). 

7.  (3o;+2)(4a?-l). 

8.  (6o;-5)(o;-l). 

9.  {2t-l){t+\).- 
10.  (o;+2)(8o;-l). 


11.  (4c-3)(3c-4f. 

12.  (ft-16)(56+l). 

13.  (a+5)(7a+l). 

14.  (2/+3)(102/-9). 

15.  (r2-2)(2r2+3). 

19.  (ar2y'-3)(2a722/+5). 

20.  (2a263_5)(3(i253_4), 

21.  3(a?+2)(4o;-l). 

25.  {x+y){2x+y).  28.   {x-iy){^x+y). 

26.  (4o;-2/)(3.T-2/).         29.  (2«i-3a)(6m+a). 


26(a-9)(3a+4).        27.  {2x+y){x-2 


30.   (20^2/— 5)(i>^+3). 


14 


ANSWERS 


[Ex.  34-35 


31.  (a;2+ 42/2)  (3£c2- 2^2). 

32.  Sa(x-2y){9x—y). 

33.  2('Sah-2c)(ab+c). 

34.  {ax+b)(x-l). 


35.  (y+2a)(2y+h). 

36.  {2z-h)(z-a). 

37.  (aa?+l)(a?+5). 

38.  2cj(a;+a)(a?— &). 


1. 

2. 

3. 

4. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 


24. 
25. 
26. 
27. 


(.17+6)  (07-6). 

(a;+10)(a;-10). 

(a;4-7)(a;-7). 

(x+8){x-8). 

{10y-7b){10y+7b). 

(9a-85)(9a+85). 

(25a;- 155)  (250?+ 155). 

(3aa7-l)(3aa7+l). 

{ixy+3){ixy-d). 

(2ay-b)(2ay+5). 

(5a7a— 4)(5a?a+4). 

(10a^-95)(10£n/+95). 

(4072;— 5c)  (4a72;+5c) . 

(13071/2;- 5a5c)  (13o7i/2;+5a5c). 

(la-lb)  (ia+ib). 
{lxy-ia)i^xy+^a). 
{i\ab-i^x)  {^\ab+^%x). 

(f-fa^2/)(l+l^^). 
(I_i^0(j5c)(l+-ijpa5c). 


Exercise    35. 

5.  (o;+12)(o;-12). 

6.  (a;-14)(o;+14). 

7.  (07+a)(07— a). 

8.  (x+n)(x—n). 


9.  (2a;+a)(2o7-a). 

10.  (3o7-a)(3ir+a). 

11.  (2o7-3a)(2o;+3a), 

12.  (4o7— 5a)(4o7+5a). 


31  /-E — ^^\  (^  -I.  ^^\ 

'  \4:b      yfUb'^yl' 

32.  l^-l\l^+l\ 

''-  {TWb-''y)(mb-'y)' 

34  i^-W^+^\ 
\9y    2xJh)y^2xJ' 

35.  / 


^_10xyz\L^  -JOxyzy 


c    /\  c 

36.  (a+5-l)(a+5+l). 

37.  {a+b—c—d)(a+b+c+d). 

38.  (x+y—a—b){x+y+a+b). 

39.  (o7— 2/— a+5)(07— 2/+a— 5). 


40.  (2o;4-22/-3a-35)(2o;+22/+3a+35). 

42  l<^+^_c+d\(a+b  j^c+d\ 

'  \a-b    c-dl\a-b^c-df' 

43  i  5(x+y)     ijx-y)  )   (  5{x+y)     4(x-y)  ) 

'  i  7(a+5)     9(a-5)  J    |  7(a+5)  "^9(a-5)  f  ' 


Ex.  36-38J  ANSWERS 

Exercise    86. 

1.  (a-b-x){a-b+x).  7.   (a+5&-l)(a+55+l). 

2.  {{+x-y){l  +  x+y),  8.   (a-h-2)(a-b+2). 

3.  (x-y-l)(x-y+l).  9.   (a-4b-2c)(a-ib+2c). 

4.  («  +  3b-3c)(a+36+2c).  10.   {l-a+dx)(l-\-aSx). 

5.  (l_aT-42/)(l+a;+%).  11.   (3£c-2a+3c)(3a?+2a-3c). 

6.  (x+2a-y)(x+2a+y).  12.  (a+6-c4-d)(a+6+c-d). 

13.  {x-2y-a-3b){x-2y+a+Sb). 

14.  (a-56— 2?w-f-3n) (a-56+2m-3n). 

15.  {x-Qa-2y-b)  (x—Qa+2y+b). 

16.  (32/-a?)(2/+3a?).  19.   (lla-46)(-a-26).  • 

17.  {Sx-4y)(x-2y).  20.   (-Sy)  (2£c).  , 

18.  (-2x-y){ix~9y).  21.   (9a+36)(lla-6). 

Exercise    37. 
1.   ix^+x+l){x^-x+l).  2.   (l-aj+2ic2)(l+a;+2a;2). 

3.   (x^-x^+l){x^-x+l){x'^+x+l). 
4.   (a2+2a6+362)(a2_2a5+362).         5.   (a2+3a+3)(a2-3a+3). 

6.  (2^2+207?/+ 5?/2)(2a;2-2.T2/+ 5^2), 

7.  (2a2+562+4aZ>)(2a2+562-4a6). 

8.  (x^-nxy-dy^){x^^'6xy-dy^). 

9.  (6a2-4a?/-5^2)(6a2+4ay-52/2). 

10.  (7a4-3a2?>+262)  (7a*+3a26+252). 

11.  (237*- 6a;22/3+ 52/6)  (2a^+6a;2?/3+52/«). 
•    12.   {8x^y^-2xy+l){8x^y^-\-2xy+l). 

13.  (a*-4a2a72/2_3a.22^)(a4+4a2iC7/2-3ic2^4). 

14.  (2ni^ii*—5ab»mn^-6a%^)  (2m^u'^+r)ab^mn^-5a^b^). 

15.  (.T6-cT8i/r3+4)(a7«+a;Vl34-4). 

16.  (oa^-2ab»c+2b*c^)  (5a^+2ab^e+2b*c^), 

Exercise  38. 

1.  (x+y){x^—xy+y^). 

2.  {x+y)(a^—oc^y+x^y^—xy^+y*). 

3.  (x+y)(x^-a^y+x*if-x^y^+x'^y*-xy^+y^). 


15 


l^  ANSWERS  [Ex.  38-39 

5.  {x+y)  {x^o-x^y+x^y^-x'^y^+afiy*-x^y^4x*y^—x^y''+x^y^-xy^+y^oy 

6.  (dx+a) {9x^-'6xa+a^).  9.  5(2a+36)(4a2-6a6+962). 

7.  {4x+^y)  {Ux^-12xy+9y^).  10.  2(Q+x)(S6-6x+x^). 

8.  (5a+6&)(35fx2-30ab+3662).  11.  3(3a;+l)(9a;2~3a?-hl). 

12.  (b+i)(b*-b^+h^-b+l). 

13.  (i+x){i—x+x'^—x^+a^—x^+x^). 

14.  (2+!c)(16-8ic+4a;2-2ic3+a?*). 

15.  (l+5a)(l-5a+25a2-125a3+625a*). 

16.  {ah+l){a^t^-ah+l). 

#     17.  (007+2)  (a4a?4-2a3ic3+4a2x2-8aa;+16). 
18.  {a+5xy)(a^—5axy+25x^y^), 

21.  0a+5c)(^a4-Ja36c+^a252c2-ia&3^34.54c4j. 

22.  (4a+i&)(256a4-16a36-}-a262-3-Va63+^^e&*). 

23.  (2x+j^y)  (Q'ix^-lQa^y+ix*y^-x^y^+ix^y*-j\xy^+-i^y^). 

24.  (a7+2/'^)(a72— ic?/2_,_2/4), 

25.  (3a;2+22/)(9a?4-6a;22/+42/2). 

26.  (x+l){x^-x+l)(x^-x^+l). 

27.  (a+6)(a*-a36+a262-a53+b4)(ai'>_a555+5io), 

28.  (1+a)  (l-a+a2-a3+a4)  (I_a6+ai0-ai5+a20). 

Exercise  39. 
1.  (x-y)(x2+xy+y^).  2.   {a-h){a^+ab+l^). 

3-  (a;-^)(a74+a?82/+a?2^2+a,^_^2/4). 

4.  (ic— y)  (x^+afiy+x^y^+y^x^+x^y^+xy^+y^). 

6.  (x-y)  {x^^+x^y+x»y'^+x^y^+x^y*+x^>y^+x^y6+o(fiy'i+x^?f+xy^+y^^). 

7.  (2a-y)(Aa^+2ay+y^).  10.  {2d^-b)(ia'^+2a%+b^). 

8.  (3a;-4a)(9x2+12a7a  +  16a2).  n.  (i_|;^)(i+|^+^a;2). 

9.  (5-7a)(25+35a+49a2).  12.  {x-l)(x*+x^+x^+x+l). 


Ex.  39-40]  ANSWERS  I7 

13.   (2-y)(lQ+8y+4y^+2y^+y*).        14.   (3-a)(81+27a+9a2+3a8+a4). 

15.  {x—2a)(x'^+2x^a+4x'^a^+8xa^+lQa'^), 

16.  (x^l)  {x^+x^+x^+oc^+x^+x+l). 

17.  (l-a)(l+a+a2+a8+a4+a5+a6). 

18.  {2-xy)  {Q4:+d2xy+16x^y^+8x^y^+4x^y^+2xY^+xf^y^). 
19.  3(aT-l)(9a?2+3aj+l).  20.  3(6-a)(36+6a+a2). 

21.  ^(x-y)(x2+xi/+7j'^). 

««    /      a\  /  .     a^a    x^a^    xa^    a\ 

22.  (^-g|(a^+-^-+^+^+_). 

23.  2(^x-y) (^i^x^+la^y+ix'y^+lxy^+y^). 

24.  2a?(fa;-2/)(fa?2+3a.^+2/'^). 

25.  a8(^_(|r2/)(a;2+a?a?/+a22/2). 

29.  (a-b-c+d)  {(a-by+  (a-b)  (c-d)  +  (c-d)^} . 

30.  (x-2y-2a-b)  {{x-22j)^+  {x-2y}^(2a+b)  +  {x-2y)^{2a+by^i- 

(x-2y){2a+b)^+{2a+by}. 

Exercise    40. 

1.  (a2+62+a6i/2)(a2+62-a6i/2). 

2.  (a2+62)(a2+52+a&i/3)(a2+62_a&i/3). 

3.  (a4+64+a262|/2)  (a4+54_a262|/2). 

4.  (a2+62)  (a8_a652_^aV-a266+68). 

5.  (a4+54+a262|/3)(a4+54_a252|/3) 

(a2+62+(^;^y^2)  (a2+62-a5i/2). 

6.  (a^.+l)(a2+l+a\/d)(a^+l-ai/3). 

7.  (ic2+l)(ir;i2-;t;io+x8-ar6+£c4-a;2+l). 

8.  (x^+l+x\/2)(x^+l—x\/2). 

9.  (a;24-l)(a;8-aj«+a3*-aj2-|-i); 

10.  (l+a:*+a'2|/2)(l+aj*-ic2|/2). 

11.  {4x*+l+2x^\/2)(4x*+l-2x^l/2), 


18  ANSWERS  [Ex.  40-41 

12.  (l+9ic8+3ir*i/2)(l+9a38-3a^]/2). 

13.  (4^2+92/2)  (ix^+gy+mxyy'S)  (4a;^+9y^-Qxy\/3). 

Exercise    41. 

1.  (a2+62)(a+5)(a-6). 

2.  (a+6) (a-6) (a2+a6+52) (a2_a?>+52). 

3.  (a-f-6)  (a-&)  (a2+?)2) (a2+52+ct?,^/2)  (a2+52_ot5-j/2). 

4.  (ci+b)  (a-b)  (a'^-a^+a^b^-ah^+b^)  (a^+a^b+a^b^+ab^+b*), 

5.  (a+b)  (a-b)  (a^+b^)  (a^+ab+b'^)  {a^-ab+b'^)  (a^+b^+ab\/d) 

(a2+52_c(5-^/3), 

6.  (a+b)  (a-b)  (a^+a^b+a'^b^+a^b^+a^^+ab^+b^) 

(a^-a^b+a'^b^-a^^+a^^-ab^+b^). 

7.  (a-^b^)(a^+b^).  9.  {a^-'l^){a^+b^){a^+b^). 

8.  (a3-64)(a3+64).  10.   (a5-62)  (a5+62.).     , 
11.  (a?2+l)(a;+l)(£c-l). 

yi2.  (a;+l)(a?-l)(aj2_^a;+l)(a-2-a;+l). 

13.  (l+x)(l-x){i+x2){l+x'^+x\/2)(l+x'2-xi/2). 

14.  (l+a?)(l— a?)(l+a;+a?2-|-£c3_i_^)(l_a;_^^.2_£t.3_^^)^ 

^15.  (1-0?)  (i+x)  (l+a-2j  (l+a?+a-2)  ( 1  -xi-x^)  (l+x^+xi/d) 

(l+x^-x\/d). 

16.  (2+a^2)  (2-0^2). 

17.  (2+a;2)(2-a;2)(2+2iC+£c2)(2-2aj+a?2). 

18.  (9-073) (9+0^).  19^   (10_a8)(10+a8). 

20.  (2a-l)(2a+l)(4a2+l)(4a2-2]/'2a+l)(4a2+2i/2a  +  l). 

21.  {Sa^+2x»){Sa^-2a^)(9a^+Axfi). 

22.  (H-2a2)  (l-2a2)  (l+2a+2a2)  (l_2a+2a2)  (l+4a*+2a2|/2) 

(l+4a*-2a2]/2). 

23.  (K2ic)(l-2aj)(i-a;+4a?2)(iH-a;H-4a?2). 

-^24.  {xy-ab){x'Y+abxy+a^b»){xy+ab)(x'^y^-abxy+a^b^). 

25.  (2+3a6)(2-3a5)(4+9a262). 

26.  (2-a)  (2+a)  (16+8a+4a2+2tt3+a4)  (16-8a+4a2-2a3+a*). 


Ex.  41-43]  ANSWERS.  19 

29.  (Factors  of  x^'+a'^)  (factors  of  aj^—a").  / 

30.  (£c»-3— 2/»+2)(cc"-8+2/»+2). 

31.  {(a+6)2+c2}(a-f-&+c)(a+6-c). 

32.  {(j;_2/)2+  (a-6)2;  {x-y+a-h)  (x-y-a+b). 

33.  {(a+b)2+c2}(a+6+c)(a+5-c). 

34.  [(2x—yy^+  (a+4b)2}  (2x—y+a+ih)  (2x-y-a-4b). 

Exercise  42. 

1.  (a+b)(c-d).               5.  (a2+i)(5_|_c).  9.  (x-l)(y-l). 

2.  (a7-7/)(a-5).               6.   (a+b)(x^+x+l).  10.  (a-2)(6-3). 

3.  (a;+2)(a+6).               7.   (2+d)(a-6+c).  11.  (£C+b)(£c+a). 

4.  (x-y)(a+4).               8.   (a;-l)(«-6).  12.   (a?-2)(a?+a). 

13.  {x—a){x+a+c).  16.   (a+6)(a7+l)(a7— 1). 

14.  (x-a)(l+x-a).  17.   (a7+l)(a:-l)(a;+3)(ic2+l). 

15.  (a7+2/)(«  +  l)(«^-«+l).  18.  (x-y)(x+y-'i). 

19.  (a-6)(£tf-l)(a.^+(r3+ic2+.T+l). 

20.  (a+1)  (a-1)  (6+1)  (5-1)  (a2+l)  (b^+1). 

21.  (07— l)(acc+5).    '         '  24.   (mn+l)  (pq+2). 

22.  (2o^+2/)(^+l)(^^-^+l).  25.   {2a-b)  (2x-y). 

23.  2fe2(a+6)(a-b).  26.  (6+(i)(6-d)(a-c)(a2+ac+c2). 

27.   {a-l)(a+l)(x-l)(x^+x^+x^+x+l). 
28.   (aH^5)(£C+l)(ic+2).  29.   {x-y)(x+2y\ 

30.  {m—q)\m+q)  (n+p)  (n'^—np+p^)  (n—p)  (n'^^+np+p^). 

31.  (a+&)  (a-J)  (x+y)  (x-y)  (x^+y^)  {x^+y^+xy\/2) {x^+y^-xy\/2). 

32.  (a-b){a+b)(a^+b^){x-y)(x2+xy+y%         ^ 

33.  a(a+6)(a;+l)(a;-l). 

34.  {x+y—z—w){x—y—z+w). 

35.  (a:+2/+'2^)(^+2/-^)(^-2/+^)(^-^+2/)- 

36.  {a-b)ia+b+cy  39.   (ic2_2/24-i)(aj2+2/2). 

37.  (a+6+2c)(a+6+3c).  40.  (x-y+z)(x^+xy-{-y^). 

38.  (a-5+c)(a-5-c).  41.  {3-x~x^)(l+x). 

42.  (a+6)(aa;+62/+c). 


20 


ANSWERS 


[Ex.  43 


1. 

2. 

3. 

4. 
13. 
14. 
15. 

20. 
21. 
22. 
23. 
24. 
28. 


(2a;+l)(ir-l). 

(Sx+2){x+l). 

(2a;-l)(aJ-2). 

(4x-l){x-h2). 

(x-i){2x-l)(x+e). 

(x-l)inx^+x+2). 

{a+2)  (a^+a+l). 


Exercise  43. 

5.  •(2a;-3)  (x-d).  9.  ^(x+2)  (x^-x-1). 

6.  (3a;+5)(a;+3).  10.  (x-l){x+S){x+5). 

7.  (Sx+2)(x-n).  11.   (x+l)(x^-2). 

8.  (x-l){x-2){x-d).  12.   (a7_l)(£c4-l)(a?2+4). 

16.  (a+3)(a2-a-l). 

17.  (a+l)(a+4)(a-5). 

18.  (a+4)(a+3)(a3+l). 


19.   (a+1) (a-1) (a2-ai/3+l) (a^+a\/2+l). 


(2x-y)  (x-y). 

{^x-y)(x+y). 

ix+y)(x-y)(x+2y). 

(x-dy){x+Sy)(x~2y). 

(a+db){a-h){a+2h). 

a(x^-Sa^)  (x^+Sa% 


-  (^-irnr- 


47. 
48. 
49. 
50. 
51. 
52. 
53. 
54. 
55. 
56. 


ah(2x-l){x+1), 
(2a+35)(2a+13b). 

(ar-7)(ar+4). 
a{x—a)(x—5a). 


35.  {y-z){x-z). 

36.  (x+'S)  (x^+6). 

.37.  (2a-2b-y-z)  (2a-2b+y+z). 

38.  (3£C-52/)(3a;-42/). 

39.  (a2-a54-62)(a2+a6+52). 

40.  {x^—ixy—y^){x^-{-4:xy-y^). 

41.  (a^3_6)(a?3+7). 

42.  {x^-5y^)  (x^+ly"^). 

44.  £c(it?-3)(l-ir)(l+a!?). 


45.  (by+l+xy){l-xy). 

46.  (x-y-z)  (x-y+z)  (x+y-z)  (x+y+z). 


(x+y)^(x-y). 

x(y-d){y+10). 

(c—b)(a+d). 

(x+d){x-2)(x^-x+Q). 

(8f+9)(9f-5). 

(a+1)  (a-1)  (5+1)  (5-1). 

(2z-M){z+U). 

-a2(3a-£c)2. 

x{x^+c)(b+ax). 

(a2_b2_c2)2. 


57.  (ic-4a4)  (a;-a252). 

58.  (£C— a— 5)(ic+2a). 

59.  (2x''-dxy-\-Sy^)  (2x2+dxy+Sy^). 

60.  (a— c)(a+c)(5+c). 

61.  (x-S)  {x^+2x+2). 

62.  (a+3)(a2-2a-l). 

63.  {x-l){x-Z)(x+b). 

64.  (a;-l)(a;-3)(ar-7). 

65.  (a;-l)(a;-2)(aj-3). 
66.(a;+2)(ic+3)(a;-3). 


Ex.  43-45] 


ANSWERS 


21 


67.  (y-5x-3z)(y-5x+dz). 

68.  (3a;»+52/")(2a7''— 32/"). 

69.  (3a-36-3)(a-6-4). 

70.  (7a  +  76+3c)(a+6-2c). 

71.  (x+3)(x-i)(x^-5). 

72.  (a-1)  (a+1)  (a-h). 

73.  (a?-a)(a;-3). 

74.  aa7(a?— 2a)  (a?— a). 

75.  {x-2)i2x^+x-7). 


76.  (5a;-7)(7ar-5). 

77.  x»(x-7){x+S). 

78.  (a7-3b-2)(ar-3[>4-2). 

79.  (.x-3_5)(a;24.7).^^ 

80.  (x-l){x-i-2)(x+S). 

81.  (a;-3)(a;2+7). 

82.  (x+l)(x^+24x-16). 

83.  (2a;+22/-2;)(5a7+52/4-6«). 

84.  (8a+8?)+9c)(3a+36-4c). 


1. 

2a262. 

2. 

3a363c2. 

3. 

15a^2/3. 

4 

8xyz^. 

5. 

a%. 

6. 

tzK 

7. 

icV- 

8. 

7a53.^2. 

9. 

Ixyz^ 

10. 

dabd^. 

11. 

xy^z. 

12. 

2a^W. 

13. 

d6xyz. 

14. 

35a3&. 

15. 

6ma2. 

16. 

5a2a?. 

17. 

7a2Z>2. 

18. 

7CC2/2. 

1. 

6a268. 

2. 

30a36*. 

3. 

8a762c8. 

85.  (a;+2-2a+y)(a;+2+2a 

-y\ 

Exercise  44. 

19.  ar32/5. 

37.  2a;-3. 

20.  2(a+6)2. 

38.  ax{x—a). 

21.  7{x-yy. 

39.  a2_^5. 

22.  4(a+£c)2. 

40.  a(a+2a;). 

23.  x-1. 

41.  1. 

24.  5(a?+l). 

42.  7m2+5m+5. 

25.  VS(x--i)^x+l). 

43.  a+h. 

26.  a;+2. 

44.  1-a;. 

27.  a:-l. 

45.  a-&. 

28.  2a;+l. 

46.*  m-2. 

29.  4. 

47.  a?— 2y. 

30.  x-1. 

48.  a2+a5i/2+52. 

31.  a;-l. 

49.  (a+l)2. 

32.  a-&. 

50.  l+ir+a;2. 

33.  a-b. 

51.  a—x. 

34.  ic. 

52.  a;-5. 

35.  a;-3. 

53.  2{x-\-y). 

36.  a;+l. 

54.  a-1. 

Exercise  45. 

4.  lQ2x^y^. 

7.  100a2m2n«. 

5.  42ic8?/822. 

8.  60^6^8. 

6.  Sia'b^c^. 

9.  144aW66. 

^2  ANSWERS  [Ex.  45-46 

10.  108x^y\  12.  3(a-6)8(a+?>)2.         14.  oc^(l+x)^l-x). 

11.  2(x+y)^  13.  2£c2(a7-l)(ii7+l).        15.  30  (a+6)2(c-d)8. 

16.  235a8a;(a-b)(2a+5)3.  23.   (2x-l)(x+2)iSx+l). 

17.  (a;+l)(£t?-l)2.  24.   (2x^-x-\0)(2x^+x-d). 

18.  (a?+l)(a?-l)(a;-2).  25.  2a7(a7+l) (072+50:4-6). 

19.  x{l-x){l+x)(l+2x).  26.  2a;2(a?+l)(a;-l)(£c2+a:+l). 

20.  ia-b)(a-\-h){2a+h).  27.   (ic+l)(a;-l)(aj+2)  (i»2+i). 

21.  (aH-2)(a+3)(a+4).  28.   (l-aT^Xl+aj+icO- 

22.  (a;+6)(a;-5)(a7-l).  29.  2x^{l+x)(l-x){x-2){x^+l). 

30.  12a;2(a7+7)(a;_2)2. 

31.  (a-&)(a+6)(a2+&2)(c[24.of5+52). 

32.  l-ic8.  37.  07(1-0?)  (l+x)(l+a;2)(10-a;). 

33.  076-1.  38.  (07-l)(a;+l)(jr+2)(a;+3). 

34.  6o;2(£c+3)(o;-l)(o;2+l).  39.  3o74(o7-l)(ic2+ir+l). 

35.  (a4-64)2.  40.  (a+b)^a-hf. 

36.  (3a-2)2(3a+2)(9a2+6a+4).  41.   (a;-3)(.T-12)(o;2-2). 

42.  (a+l)(a4-2)(a-l)2(a2-2a+3). 

43.  (6a3+a2-5a-^)(3a2+a-2). 

44.  (6o78-7o72z/-2a;y2)(3^2_,_^2/-4^2),  , 

45.  (07+1) (07-1)  (x+2)  (x-2){x-r^)  {x-S). 

46.  (07+2) (207-1)  (307+1). 

47.  (2a+l)2(2a;-l)2.  49.   (dx-^)(2x+7)(ix-n). 

48.  (a-b)(x-y)K  50.   (4a;+l)(2a;+7)(3a?-8). 

Exercise  46. 

1  «  '7x^  ab-2 
'    b'                                 6-   2p- 

2  2a2  5o^ 
^'    362-                               7-    "7^- 

3  528  __3266c 

4  — .  9     -^^ 
6.   f .                                10.   ^|. 

»  07+1 


Ex.  46-48] 


ANSWERS 


23 


16. 
17. 
18. 
19. 
20. 
21. 


x+2 
x^  ' 

x-y 

xy  ' 

x+1 

x^+x+1' 

x—d 

X  +  4:' 


X' 


■x+1 


X^  —  0(^+X^—X  +  l' 


3.r+ 


x^' 


1 
2 
3. 

10.  3a +86+ 


2 


X—1  + 


ix 
6x2' 


x-V 

1062 

3a -26* 


_3_ 

6£C2' 


22. 


23. 


24. 


27. 


x+2 
3x+i' 

1+x+x^ 
x+x^+a^+oc^' 

x'^+2xy+y^ 
x'^+xy-\-2y^' 

1+x^ 
1-x^' 

l-g 
2+a* 

x+1 

x+2' 

Exercise  47. 


28 


4  —  3? 

2  +  07' 

x^+Zxy+y^ 
^^-   x^-2xy-iy^' 

b+c—a 


30 


31. 


b—c+a 

m-2 
m+9' 

c—d 
c+d' 

x+2. 
x-d' 


4.   £c2+2a?+l. 


7.    2a;+l  + 


2a:2_i 


5.   x+y+ 


x-y 


6.   x^-dx—2+ 


8X  +  4: 


2v^ 
^    x+y 

a'2-2.r+4+ 


x+2 


11.   x^+2x:^+'lx+8+ 


32 
a;-2* 


12.  x^—xy+y'^ 


hx 
6a;2- 


2y3 
i»+2/' 

Exercise    48. 

8a* 


9a3 


20a8a; 


24a3ic3'       24a3.T3' 
.r2— ,r  ic^+a? 


1' 


1' 


6a?3 
24a«a^* 

1 
a;2-l- 


3. 


4. 


18&4 


3c3 


12a262c' 
xyz' 


6£C+3 


5a? +5 


12a262c'       I2a262c* 

£C2  ?/2 

(ri/2;'  a?^/^;' 


707+14 


(a;+l)(a;+2)(2a;+l)'    (a7+l)(a;+2)(2ic+l)'    (a?+l)(a;+2)(2a:+l) 

(2a;+l)(a;-5)  (l-a?)(a;-1)  x(x+l) 

'    {x+l){x-l){x-5y    {x+l){x-l){x-5y    {x+l){x-i){x-5)' 


(a+b)(a'^+b-2) 

a^-b^ 
2(a;+l)a 
(a;+l)*' 


2(a-b)(ag+b2) 
2a;2 


{a—b){a+b) 


x{x+l) 

{x+1)*^  {x+iy' 


24 

10. 

11. 

12. 
13. 
14. 
15. 
16. 

1. 
2. 
3. 
4. 
5. 

17. 
18. 
19. 
26. 
28. 


ANSWERS 


[Ex.  48-49 


xir 


a;(a;2_j,y2)     .r^faT+y)     ^^___ 
y{x^-y^y  y{x^-y^y  y{x'-y'^)' 
6a?(a?+?/)     -3?/(a?+y)  x^ 

x(y-b)(z-c)  y(x-a)(z-c)  z(x-a)(y-b) 


-2?/2 


{x-a}{y-b){z-cy    {x-a){y-h){z-cy    {x-a){y-b){z-cy 


x^-9 


x^-1 


(x-l)(x-2)(x-dy    (x-l){x-2)(x-'dy 

2a^+ia  4ac+12c 

■(a+2)(a-2)(a+3)'    (tt+2)(a-2)(a+3)* 

b—c c—a 

{a—b){a—c){b—c)'    {a—b){a—c){b—c)'    (a—b)(a—c)(b—c)' 


n-b 


y'^-z,^ 


z'^-x'^ 


x^-y"^ 


{x-y){x-z){y-zy    {x-y){x-z)\y-zy   {x-y){x-z){y-zy 
Exercise  49. 

-      4.^2+37  +  1 


7.-r-2 
-  2a;2  • 

abc 

Ux-j-9 
lOa^  * 

x+y+z 
xyz 

Ax 
x^-1' 


a+2h 

10^2+20? 

x'^-\     ' 

1 
2+y' 


7. 

8. 

9. 

10. 


a^—x 

3a;«-2a;2+8a;-2 
2oc^—5x^+2x  ' 
4a2 

a2-a;2* 

x+1 

X 

2x+x^ 
1+x 


11. 
12. 
13. 
14. 
15. 


16. 


20. 
21. 


(a;+4)(2a;-l)(3a;+l)' 

a;2_i- 
2&2 


22. 


4bcd + 6acd —Sabd— 2abc 
ASabcd 

y^ + xy^ + Sxh/ — x* 

x^^  •     ^^• 


2xy—2y 
x^—xy^' 

27.  0. 

oc^—x'^+dx+l 
x^~l 


24. 


25. 


30. 


1-x 
—x^+x^—2x 

1-£C2  • 

5— a;— 5a?2 

l-X'2      * 

4ab 
cC^-b^' 

-4 

{x+\){x~'d){x+by 


2a?2— 6a?— 14 

a;2-2a?-8  * 

a^ 


x-r 


(a—b){a—c) 


4ax+4bx—4:ah 
452- a;2       • 


Ex. 

31. 

35. 
37. 
38. 


49-51] 


x^-l 


32 


ANSWERS 


26 


24-8a;2 


34. 


1. 
5. 

9. 
10. 
11. 

12. 

13. 

14. 

15. 

16. 

1. 
2. 
3. 
4. 


14a;2+130 

£C4-26£C2+25" 

4.T3+2a;2+4a;-5 


ad 


x+y 
x+2y 


x^+xy+y^ 

x^+ocy+y^ 
x^—xy+y^' 

a+1 
a+o 

x+y 

x^ 

h^     ' 
^x'^-\-^xy 

b-2+3ab+9a2 
68 

3a^ 
bxy 

1 
3a -36* 

x^—xy+y^ 


x^—5x'^y^+4y^' 

2bc^-2bd^+2a^-2bm 
a^c^-b^c'^-a'^d'i+b^d^  ' 

^^    2iic*-2a*+2a^xi-2a^ 


39. 
40. 


y  ^ 

a^xz' 

2y+S 
x-1' 

17. 


2x^-x^ 
x^-1  ' 

x(x+y) 
x-y 

Exercise  50. 
3.   1. 

7.     07-1, 


x^—x^a^—x^+a'^ 

a^+a%+ab^+b^ 


1 


x-1 

18.  a+b. 

19.  x+y. 

20.  x^—x^y+xy^. 
a 


21. 
22. 
23. 
24. 

5. 
6. 
7. 


a—b 

2(x-y) 
x+y 

x+y-S 

X 

x-y 
x+y 

Exercise    51. 

x'^+4xy+4y^ 
2x'^+xy—'Sy^' 

ab^-a^ 


x+\ 
x+^' 

8.   1. 


41. 
42. 


a2-62 


25. 
26. 
27. 


1 

2+x 


4.  a«6. 

8.  a^-x'^+x-\. 

2ofi—x'^+Ax-2 
a;2-4 

225a2c2-16a?* 
25c2 

x^ 

x+2' 

oc^—Ax 


a?^- 9 

29.  2x. 

30.  6. 

31.  6a;+9. 

32.  a;2+a;-4. 

33.  4a2. 

x—2 


10. 


a;'^+9a;+14' 

(a-6)2(CT2+6g) 
(rt-6)2+l     • 


^^'      6xy^    ' 


12. 


x+x^ 
1-x' 


26 


ANSWERS 


[Ex.  52-55 


Exercise    52. 


U 

X 

x-\ 
x+V 

1+237 

l+a7+a;2* 


13. 


a 

6 

a^+4a72+7a7+6 

.T3+5x2+9a7+5" 

7. 

8. 

a2+62 
a2-62- 

a2-a+l 

10. 


CT-4 

a— 5* 

?l2 


11.  3a72-ir2/-32/2. 


2a- 1 


14. 


307+3* 


Exercise 

53. 

1. 1. 

6. 

8. 

11.  ^. 

16.  I, 

21.  3. 

2.  -2. 

7. 

-6. 

12.  15^. 

17.  16. 

22.  If. 

3.  A. 

8. 

0. 

13.  8. 

18.  -3 

23.  If. 

4.  2. 

9. 

1- 

14.  -|. 

19.  V. 

24.  12. 

5.  2. 

10. 

i. 

15.  -37. 
Exercise 

54. 

20.  f. 

25.  2. 

1.  6. 

7. 

I 

13.  4. 

- 

19.  4. 

25.  8. 

2.  6. 

8. 

h 

14.  -H. 

20.  1. 

26.  4. 

3.  -f. 

9. 

4. 

15.  19. 

21.  |. 

27.  -2. 

4.  5. 

10. 

6f. 

16.  f. 

22.  \. 

28.  -10. 

5.   -f. 

11. 

1. 

17.  13. 

23.  ^\. 

29.  -S^. 

6.  1. 

12. 

15.  . 

31.  3 

18.  -7. 
Exercise 

32 
55. 

24.  13. 

30.  ^. 

1    ^+cy 

'      2a  ' 

6. 
7. 

1. 
2a- 1 

-     11. 

a+b—c  ' 

2.  y-2. 

3.  1. 

8. 

4-a* 
—a. 

12. 

a(x+S)+h(x-S) 
x-1 

4    c(a;-l)- 

07- 

-a(a;+3) 

9. 

2ax-h 
c 

13. 

4t+l 
t+2' 

5.   3a;. 

,^ 

10. 

0. 

14. 

-t. 

Ex.  55-56]  ANSWERS  2Y 

15.    1  19.    ?f.  23.   ^. 


r  '  v^' 


c 


16.^.  20.  f?.  24.  £^'. 

17.  ^.  21.  ?^.  25.  |. 

18.  i!:.  22.  E±D:,  26  .:^ 

27.  f(F_32).                   28. 


rr 
r+r' 


Exercise    56. 

1.  $10.  6.  16  and  20.  11.  96. 

2.  12  and  38.  7.  -24.  12.  26. 

3.  -7  and  8.  8.  3  and  25.  13.  17  and  18. 
4.-3  and  12.  9.  25  and  36.  14.  j%. 

5.  24.  10.  27.  15.  9. 

jg  16  yrs.=:  John's  age.  21.  293^  mi.  from  station. 

'  10  yrs.=  James' age. 

.«  .«  ,  ..  22.  g^Vmin. 

17.  10  yrs.  and  40  yrs.  ^ 

18.  $125.  ^^'  ^^' 

19.  $72000.  24.  10  rods  by  16  rods. 

20.  45  dimes  ;  6  quarters.  25.  6. 

26.  1 J  mi.  per  hr.  10  mi.  3^  mi.  per  hr. 

27.  -36.  31.  $350. 

28.  A,  $120;  B,  $160;   C,  $80.  32.  8%. 

29.  11  and  12.  33.  Silk  $1.10;  linen  $.55. 

30.  -2.  34.  18  ft.  by  20  ft. 

35.  In  8|  hrs. ;  21 J  mi.  from  starting  point  of  first  pedestrian. 

36.  21  y\  min.  after  4  o'clock.  41.  60. 

37.  5y\  min.  before  5  o'clock.      42.  $3750  and  $2500. 

38.  15  min.  43.  12  lbs.  iron  ;  60  lbs.  lead. 

39.  8f  da.  44.  25  oz. 

40.  6  da.  45.  246. " 


28.                                              ANSWERS  [Ex.  57-60 

Exercise  57. 

1.  x=2,  y=S.                  9.  a=5,  x-Q.  17.  x=7,  y=10. 

2.  x=-l,y=^.               10.y=5,b=7.  16.  x=^2,  y=16. 
Z.  x=5,y=L                  11.  a=rf|,5=-if.  19.  x=l,  y=i. 

4.  x=-2,  y=-S.           12.  07=36,  y=d6.  20.  x=-d,  y=ll. 

5.  x=-h  y=h              13.  a=^,  b=-'^.  21.  x=iO,  y=lb, 

6.  a- -2,  b=-d.           14.  ir=fff,  2/=fA.  22.  x-1,  y=A. 

7.  a7=4,  a=ii.                  15.  jp=2,  g=5.  23.  x=a—b,  y=a—b. 

8.  a=3,  6=7.                  16.  .t=12,  ^=11.  24.  aj=:a,  y=b. 

25.  a;=a,  2/=|- 

1.  a;=13,  y=17.                6.  a=35,  5=20.  11.  a=12,  6=-13. 

2.  x=4,  y=-5.                7.  a=7,  6=9.  12.  6=6,  2^=18. 
Z.  x=il,y=^j\\              8.  a;=15,  a=8.  13.  m=Yi^,  ?i=ff. 

4.  a;=8,  2/=-2.                9.  a=-l,  2/-3.  14.  a;=7,  ?/=-2. 

5.  x=2,  y=l.                  10.  aj=14,  y=zU.  15.  a?=1.8,  2/=1.4. 

16.  07=6,  2/=12, 

17.  a=3,  6=-7. 

18    x=^^^-tl^      V- 

2n+m'     ^    2n+m 

Exercise  59. 

1.  a=17,  6=13.                7.  a;=10,  y=2A,  13.  x=7,  y=-2. 

2.  a=-3,  £c=-7.            8'.  aj=l,  2/=-f.  14.  a? =5,  2/= 7. 

3.  a7=9,  2/=2.              -^.  a;=H,  2,=,?,.  15.  a?=2,  2^=3. 

4.  m=  — 11,  n=7.           10.  x=l,y=—\.  16.  a;=6,  2/=4. 

5.  £c=8, 2/=l.                  11.  a=5, 2/=-3.  17.  a;=9,  2/=-3. 

6.  a;=-V/-,  2/=/t-            12.  a7=2,  2/=8.  18.  a7=12,  2/= -3. 

19.  0?=^,  2/=tV                       20,  aj=12,  2/=3. 

Exercise    60. 

1.  a?=7,  2^=3.                   4.  a?=i,  y=l.  7.  a?=20,  2/=32. 

2.  a?=7,  2/=4.                   5.  a=4,  2/=3.  8.  £c=-l,  2/=3. 

3.  a;=5,  2/=7.                   6.  a=-2,  6=-3.  9.  a?=ll,  y=-4. 


Ex 

.  60-64] 

ANSWERS 

( 

10. 
11. 

x=5,  y=-5. 
x=8,  y=l. 

14. 
15. 

a?=20,  y=12. 
a;=12,  y=10. 

18.  x-   ^  ^,y-   ^  ^. 
a+b '^     a+b 

12. 

a=—%\h=—'l. 

16. 

x=67,  y=10d. 

19.  x=a^+b^,  y=ab. 

13. 

00  =  4:,  2/=f. 

17. 

a=10,  x=8. 

20.  x=a^,  y=b. 

21.  x=2p,  y=iq. 

22.  x 
Exercise    61. 

=18a-246,  y=SQb-2ia. 

1. 

x=u  y=2. 

4. 

a=3,  h=4. 

7.  x=2,  y=7. 

2. 

x=-2,  2/-3. 

5. 

x=S,  y-2. 

8.  a=l  b=h 

3. 

x=h  y=h 

6. 

a;=:5,  2/=2. 

9.x=^^^,x=-^, 

10.  x=l, 

2/=l. 

11.  07=10,  y=5. 

29 


Exercise    62. 

1.  x=l,  2/=3,  z=^.  7.  07=4,  2/=5,  5r=6. 

2.  ic:=6,  2/=l,  z=2.  8.  a;=20,  2/=10,  ;2;=30. 

3.  x=l,  y=h  z=h  9-  a?=l,  2/-7,  2;= -4. 

4.  0?=^,  y=l  z=i.  10.  p=2,  3=3,  r=4. 

5.  .T=-5,  2/=8,  2;=-9.  11.  x=l,  y=2,  z=S,  w=:4. 

6.  x^-l,  2/=8,  ;s=l.  12.  p--=2,  q=-l,  r=-3,  s=5. 

Exercise    64. 

1.  21  and  8.  9.  Father  35  ;  son  10. 

2.  15  and  -6.  10.  36  and  27. 

3.  3i  ft.  and  8i  ft.  11.  22  and  16. 

4.  96  and  24.  12.  26  and  8. 

5.  Orange  4  cents  ;  peach  1  cent.  13.  |. 

6.  A's  age  36  ;  B's  age  21.  14.  j\, 

7.  4  cows  ;  24  hogs.  15.  18  and  40. 

8.  5  dimes  ;  20  nickels.  16.  48  and  8. 

17.  10  persons  ;  $16. 

18.  A,  $125  ;  B,  $250  ;  C,  $200  ;  D,  $450. 

19.  54. 

20.  Carriage,  $175  ;  harness,  $25. 

21.  12  doz.  at  12  cents  per  doz. ;  8  doz.  at  4  for  5  cents. 

22.  $575,  $1250,  $1825. 


30  ANSWERS  [Ex.  64-65 

23.  6  men  ;  $2. 

24.  $750 ;  6%. 

25.  8.5  in.  first  year  ;  10  in.  second  year. 

26.  A*s,  2|  mi.  per  hr.  ;  B's,  3  mi.  per  hr. 

27.  24  mi. 

28.  45  mi.  per  hr.  ;  55  mi.  per  hr. 

29.  A's,  6  yds.  per  sec. ;  B's,  4  yds.  per  sec. 

30.  8  hrs. 

31.  2  mi.  per  hr. 

32.  Current  2|  mi.  per  hr. ;  crew  4  mi.  per  hr. 

33.  90  ft.  per  sec,  and  60  ft.  per  sec. 

34.  360  revolutions  per  min.,  and  540  revolutions  per  min. 
.35.  A,  30  days  ;  B,  24  days. 

36.  A,  68f  days  ;  B,  160  days  ;  C,  240  days. 

37.  Man  $2.75  per  day  ;  boy  $1.25  per  day. 

38.  A,  $23,040  ;  B,  $7,680. 

39.  35  hogs  ;  60  days. 

40.  Alt.,  9  in.  ;  base,  12  in. 

41.  Velocity  in  still  air  350  yds.  per  sec.  ;  velocity  of  wind  6.5  yds.  per 

sec. 

42.  One  in  16  min.  ;  other  in  20  min. 

43.  $1.25  and  $1.65. 

44.  6yx  from  first ;  d^^  from  second. 

45.  4  from  first ;  3  from  second. 

46.  42  lbs.  tin  ;  14  lbs.  zinc. 

47.  8j%  oz.  first ;  4|^  oz.  second  ;    2}f  oz.  third. 

48.  432. 

49.  24,  60,  120. 

50.  Length  30  rods  ;  width  20  rods  ;  area  600  sq.  rods. 

51.  384. 

Exercise    65. 
1-  2l/2.  3.  _2^;3;  5.  6l/£ 

2.  51^5.  4.  3i>?.  6.  6i/3. 


Ex.  65-67] 

ANSWERS 

7.  5i/4. 

8.  42i/2. 

9.  soil's. 
10.  21^102. 

18.  2ah-\/2a-h.               28 

19.  a2v^2. 

20.  iy^. 

21.  ii/6.                           3°- 

^V9c.,2. 

{a—x)\/  a—x. 
(^+2/)l/4a. 

11.  91^5. 

12.  Sxyi/Sy. 

13.  ia^b^i/dab. 

22.  ii/is". 

23.  ^^i^ii: 

24.  i^/^. 

25.  .-^al^^- 

31. 
32. 
33. 

3     / — r 

■1 

14.  2axy\/2xy^. 

2^>/^-4^C. 

15.  -5ir22/4y^5a.2. 

16.  2x»y^y. 

17.  a'^y^i/Zay'^. 

2/^ 

26.  ^V'a^bcK                  34. 

27.  4^^'2'-                     35. 

-i 

|l^l+2a;+2a^. 

Exercise    66. 

1.  5i/2. 

3.  1/5. 

6. 

4V3. 

2.  21/3: 

4.  5l/6. 

6. 

-4/5. 

7.  Ii'/e: 

14.  fl/2-|/3 

8.   (3b2+2a6-a2)^^ 

15.   (3-2a+4a7)i^^. 

9.  (3a-2a25+262)|/26r 

16.  VV3. 

10.  51/  3a2. 

17.   -I1/5: 

11.   {x+2y=2)\/y. 

18.  -/^v^i: 

12.   (2a+3&)i/a6c-a6|/c. 

19.  2|/5H-6-S 

ii;^4-: 

13.  {x'^-xy)y'x^-^yH/xy\ 

20.  3v  2+11/2: 

Exercise    67. 

1.  1^08,  v^a*r 

4.  v^343,  ^/144,  1^64. 

2.  y^^  k'^,  I^^ 

5.  i/a8>6i«,  y'a^bis,  j/  a^b*. 

3.  F  5^  |^2«;  J^P5; 

6.  1/5. 

31 


32 

ANSWERS 

[Ex.  67- 

7.  i/48. 

-  ^i- 

19.  51^25920. 

8.   1/ iol 

20.  i/a^-b^. 

9.  1/192. 

-  i/i- 

21.   (a;-?/)  i/a?+2/- 

^ 

22.  a+b. 

10.  y'567. 

16.  21^3. 

23.  -1. 

11.  l/f. 

17.  61^500. 

24.  21/2; 

12.  i/|. 

25.  4. 

13.  \/h' 

18.  1^55296. 

26.  5. 

27.  6+2i/2+2i/3+2l/6! 

28.  ai/2a6. 

31.  c2i^3a*62. 

32.  ^]/a6. 

aa'2    / 

1 

29.  xy\/^' 

OK                      ^                ^     /     ,0          J  0 

^^-    (a+6)2l/"^-^-- 

30.  a86ci/3c. 

33-    SaW^b^' 
Exercise    68. 

36.   ^i^2a2^2. 

1.   fl/3. 

3.   Uj/U-l/2i). 

5.   4|/3. 

2.   ii/l5. 

4.   3v/2. 

6.    5i/3-5i/2. 

7.  3+1/6-1/15-1/10. 

8.  3+i/T4-v^6-fi/31. 

9.  1/15-4. 


13.  l+|ir— |i/a;'^+3£C. 

14.  l+il/2-il/6"-Ji/3: 
15-   A-Ai/iH-tVi/S-tWS. 

16.  |l/3+tl/'2-il/7-il/42. 

17.  i+ll/6-^l/l5. 


10.  b-i/b2-a2. 

11.  l+2a;2+2a7i/l+a;2, 

12.  ^+ii/;i^z^ 

13     (x+  l/xy+  -\/xz)  (x+y—z—2\/xyy 
x^+y^+z^—2xy—2xz—2yz 

19    n—b+j/ac—  }/hc  )(a+ h—c—2\/ob) 


1.   21/2. 
3.  a7i/4a72/^. 


a2+6'2+c2-2a6-2ac-2bc 
Exercise  69. 

3.  125a?V8^. 

4.  64a666. 


5.  a«. 

6.  a^b^y'a^. 


Ex.  69-71]  ANSWERS 

7.  64a66?/5.  5  __  ^  3/- 

•^  12.  1/4.  16.    2x2 1/2. 

8.  243ari'2/24.  .  V  '*.                                      ^    V 

9.  256(a2-62)2.  13.  i>5^.  1^'    l^"'-^. 
10-    V^S.  14.  v^3H^2. 


sa 


18.    |>2a;2. 


19.   7.T3aV2a. 
11.    1^2.  15.   2.  „p- 

•^  20.    \/a. 

Exercise   70. 

'•    '^^  ''    ^Y^  13.    (a+6)^— 

2.  10i/-l.  8.    7i/2l/-l.  

3.  25,/=T.  9.   aV:^  ''•    (^-3^2/)>/-l. 
4     ii/ZT                        10-   4a?3?/|/_i.                  15.   7|/-1. 

5.  ii/-i.  ^^-  7;^i/-i-  16.  121/-1. 

6.  fi/3T.  12.   9(a;-2/)2|/ZT.  17.   2y^, 

18.  (2i/2+|/10+i/7)|/-:^.  22.'  16+3i/=T. 

19.  5a2|/ZT.  23.   2a+26. 

20.  (2x^y-h6x^y^)i/~.  24.    2|/::6. 

21.  Sx+y}/'^.  25.   2a;-5?/i/^. 

Exercise  71. 

1-  -^^-  4.  -24|/'^.  7.  a363. 

2.  -12.  5.  420.  8.  -2x^y2, 

3.  -70.  6.  lOSOj/^.  9.  5. 

10.  6+i/6+2i/^-3i/^. 
11-  1-  12.  0.  13.  -10i/l0-6i/5+5i/6+3i/3. 

^^*  ^~"'  19.   -7|/^.  24.  i]/IT. 


20.  -i/H^.  25.  ^. 

21.  2.  26.  |, 


15.  y^-x^. 

16.  -5v^^. 

17.  -fi/'^.  22.  |.  27.  fic8 

18.  -5i/^.  23.  2.  28.  -|v/^. 

29.  *-fl/^.  30.  i+iv/^s+ii/iTs-ii/e. 


34 

ANSWERS 

1 

81.  f- 

-fv/-5. 

32.  f +1]/ 

-3. 

33. 

i- 

tW-5' 

34. 

.  -^■^/To-il/15-il/i4-il/21. 

35. 

x^- 

-y+2xi/-y 
x^+y. 

^ 

Exercise  72. 

1.  ±2. 

11. 

±2i/-l. 

20. 

±5. 

2.  ±5. 

3.  ±13. 

4.  ±7. 

12. 
13. 

21. 
22. 

±il/30. 

±ii/39: 

5.    ±|l/30. 

14. 

±iV2. 

23. 

±ll/55. 

6.   ±il/30. 

15. 

±Ai/-^09. 

24. 

0. 

7.   ±25. 

16. 

±l/-3. 

25. 

±1. 

8.    ±|. 

17. 

±3. 

26. 

±V/-14. 

9.   ±>/7. 

18. 

±7. 

27. 

±f. 

10.   ±2. 

19. 

±1. 
Exercise  73. 

28. 

±2. 

1.  3,  4. 

14. 

-hh 

27. 

3,^. 

2.  -5,2. 

15. 

11,11. 

28. 

-2,5. 

3.  -1,  -7. 

16. 

-h  -h 

29. 

11,  12. 

4.  -6,9. 

17. 

11,-7. 

30. 

-10,  f.. 

5.  2,  i. 

6.  -3,  i. 

7.  -2,1. 

8.  1,  -h 

18. 
19. 
20. 
21. 

5,6. 
-2,5. 

3,  -V-. 
\S  -2. 

31. 
32. 
33. 

3,-|. 
4,9. 

9.  h  -f. 
10.  2,  -i. 

22. 
23. 

2,  -V-. 

34. 
35. 

—  1,  6. 
-1,  -9. 

11.  -2,  i. 

24. 

4,^. 

36. 

28,  -20. 

12.  -2,  1. 

25. 

i,i. 

37. 

h  -v- 

13.  4,  7. 

26. 

6,  -V-. 

38. 

5,-f. 

[Ex.  71-73 


Ex.  74-75] 

ANSWERS 
Exercise   74. 

1.  2,  4. 

17. 

1,  -f. 

33.  7,  f. 

2.  -6,2. 

18. 

tV±tVv"141. 

34.    ±5. 

3.   -4,  -10. 

19. 

-i±i|/-71. 

35.  4,  -W-. 

4.  -3,2. 

20. 

tV±tVi/13. 

36.  -i±il/5. 

5.  6,  5. 

21. 

_4±|/-5. 

37.  -|±^|/185. 

6.   -7,  2. 

22. 

-3±i/-2. 

38.  i±ii/37; 

7.  9,  -1. 

23. 

i±T/3. 

39.  2±ii/3. 

8.  11,  -2. 

^^24. 

-3±4i/-l. 

40.  3,  -|. 

9.  -3,  18. 

25. 

14,  -8. 

41.  4,  11. 

10.   -2,  -10. 

26. 

3,  5. 

42.  3,  -V. 

11.  h  -1. 

27. 

-5,  -13. 

43.  -3,  5. 

12.  3,  -h 

28. 

±4. 

44.  0,  4. 

13.  h  f. 

29. 

4,-3. 

45.  |f±xVl/273. 

14.   -i,  3. 

30. 

±1. 

46.   ±4i/2. 

15.  f.  |. 

31. 

5,7. 

47.  2±ii/3: 

16.   -i  7. 

32. 

7,  -V-. 
Exercise  75. 

48.  W±^Vl/209. 

1.  -2,5. 

16. 

f±il/-ll. 

31.  6,  -1. 

2.   -3,  -7. 

17. 

tV±tV1/337. 

32.  4,  -|. 

3.   -2,  f. 

18. 

f±il/57. 

33.  0,  1. 

4.  i,  -2. 

19. 

if,  -1- 

34.  0,  h 

"^5.   -l,f. 

20. 

-f±il/l3. 

35.  l±3l/^l.  . 

6.  7,  |. 

21. 

-i±il/l7. 

36.  7,  -V-. 

7.  1,  -f. 

^22. 

l±3i/-l. 

37-  1,  -h 

8.  f ,  |. 

23. 

-i±i|/129. 

38.  3,  -V-. 

9-  I  |. 

24. 

2±i/5. 

39.  1,  1. 

10.  8,  h 

25. 

f±il/-7. 

40.  -i±il/5. 

11.   -|±il/-23. 

26. 

-l±2|/2. 

41.  0,  1. 

12.  i±i^-34. 

27. 

3, -i. 

42.  7,  V-. 

13.  A±3W-1'^9. 

28. 

-i±il/-3. 

43.  0,  -5. 

14.  4±2i/3 

29. 

3,  -V-. 

44.  h  -2. 

15.  2,  f. 

30. 

l|±iV|/l33. 

45.  3,  |. 

35 


36 

ANSWERS 

[Ex.  75-78 

46.  3,  |.                   48. 

3, 

-5.            50.  4,  -1. 

52.  4,  -|. 

47.  -¥,  h              49. 

4, 

-  4.             51.  2,  -h 
Exercise  76. 

53.  5,  -|. 

1.  ic2_^_3o^o. 

4.  2r)x^+30x+Q=0. 

7. 

x''+10x+24:=0. 

2.  x^-5x-U=0. 

5.  Gx'^+x-2=0. 

8. 

12a?2+25a?+12r=0. 

3.  16ir2_s.^+i::^0. 

6.  6ir2-7a?-10rr0. 

9. 

4a?2_7a;4-3z=0. 

10.  8^2+6j;c4.1-0.  11.  Positive. 

12.  One  positive  and  one  negative  in  each. 


13.  -A 


±|/^. 


±a. 


14.   ±  16.  15.  (a)  c  not  greater  than 

Exercise  77. 


(^)c=i. 


3.  i&±  11/62+16. 

4.  a7=:3a  or  —^a  ;  a=^a7or  —3a;. 

5.  x=a  or  2a  ;  a=x  or  4^x. 
6.  ?>=:£c  or  — iT— 2a  ;  x=b  or  —5— 2a. 


7.  — ir±i-i/r2— 4.9. 

^      1_ 
■2m'^2?^T 


8-   -9^±9^l/^'-4mf. 


).  X, 


10. 
11. 
12. 


13. 
14. 
15. 


ID 


mna 


16.    ± 


17. 


■v±\/v^+2c 


i8.5=±V|?^;'=±i|/|';-4|/f. 


Exercise    78. 

1.  15  and  22. 

3.  42  and  8. 

2.  10  and  21. 

4.  17  and  33. 

Ex.  78-79]  ANSWERS  37 

5.  76  and  77  ;  or  -77  and  -76.  8.  /^. 

6.  12  and  13  ;  or  -13  and  -12.  9.  16  or  -8. 

7.  4  and  14  ;  or  —14  and  —4.  10.  Either  3  or  4. 

11.  8  or  -  V-. 

12.  2  in.  and  5  in.;  or  4  in.  and  7  in. 

13.  500  ft.  by  596  ft.  18.  1  inch. 

14.  12  in.  and  16  in.  19.  2  ft. 

15.  5  ft.  20.  6  in.  by  12  in. 

16.  25  yds.  and  39  yds.  21.  32  rods  by  60  rods. 

17.  36  sq.  in.  22.  5  hrs. 

23.  30  mi.  per  hr.  and  40  mi.  per  hr. 

24.  8  and  12;  or  -12  and  -8.  32.  100  ft. 

25.  9  and  24.  33.  35  feet. 

26.  6.  34.  10  in.,  8  in.,  6  in. 

27.  2  mi.  per  hr.  35.  24  min.  and  18  min. 

36.  15f  min. 

37.  12  days. 

38.  15  min. 

39.  A,  8  days  ;  B,  10  days. 

40.  10  and  14,  or  -60  and  84. 

41.  45  mi.  per  hr.;  and  30  mi.  per  hr. 

45.  64  sq.  in.  48.  12  inches. 

46.  4  inches.  49.  84. 

47.  14  inches.  50.  4  ft.  by  8  ft. 
51.  20 A.                               52.  529  sq.  yards  ;  4  yards. 

Exercise   79. 

1.  2,  -2,  2i/^,  -2i/~.  7.    i/S,  -1/^,  1/2,  -1/2. 

2.  1/5,  -V/'S,  1/^,  -l/^.  8-    1»  -1.  l/2,  -V2. 

. 9.    2,  -2,  3,  -3. 

3.  2,  -2,  il/-6,  -il/-6.  ^0     ^^  _^;  ^^^  _^ 

4.  1,-1,-2.  jj     -3±i/l0,  -f±ii/29. 

^'  ^'  ^'  -^'  12  -^±y^  -i±i/5 

6.    1,  3,  -2.  •  2         '  2"      • 


28.  15doz.; 

18  cents. 

29.  12. 

30.  $500. 

31.  42. 

40. 
4.1 

42.  2ihrs. 

*i, 

43.  12. 

44.  1. 

38  ANSWERS  [Ex.  79-80 

13  ^±V~    1±V~  17.    1,     ZllAl^. 

14.  1  ±  1/2,  1  ±  1/2.  ^«-    3'  -^'  ^^  ^'  -^^'     _ 

15.  1,  -1,  l/-3,  -l/-3.  19-    1»    -1>    2 '    T) . 


16.   1,    -2,  zl±V^.  20.    l/^±/-^,    -V2±i/-2 

21.  ±2,  ±2i/~  l/2±l/^,  -l/2±i/^. 

22.  ±  i/2,  ±  |/^,  1  ±  l/^,  -1  ±  l/^. 


23. 
24. 

±l/-l,    '^'%^ 

'-1     ~ 

-l/3±i/- 
2 

-1^ 

-|±iT^34+2i/- 

-15;  - 

|±^l/34- 

•21/ 

-n 

1. 

25. 

l±l/-7    l±3i/- 
2       '          2 

EI. 

26. 

2,   4,    n! 

r±v^l7 

2        • 

28. 

-l±l/^ 

-     - 

-i± 

2 

VI 

27. 

1,   1,   — 

3±l/5 
3       • 

29. 

1,   1,  1± 

^ 

zl. 

Exercise  80. 

1. 

3. 

13.    144. 

25. 

None. 

37. 

9. 

2. 
3. 
4. 
5. 
6. 
7. 

5. 

1. 
4. 
7. 

-5. 
4,  -1. 

14.  16. 

15.  9. 

16.  -|. 

17.  7. 

18.  -12. 

19.  4. 

26. 
27. 
28. 
29. 
30. 
31. 

8. 

12. 

4. 

-1. 

I 

38. 

39. 
40. 
41. 
42. 
43 

0   «'-^* 

^'      2a2  • 
13. 

1- 
16. 

None. 
0. 

8. 

-2. 

20.   49. 

32. 

None. 

44. 

2. 

9. 
10. 
11. 

-8,  40. 

-4. 

3,2. 

21.  0. 

22.  None. 

23.  81. 

33. 
34. 
35. 

-h 
7. 

i. 

45. 
46. 

47. 

0,-3. 

7. 
1. 

12. 

5,  A. 

24.   4. 

36. 

i. 

48. 

1. 

Ex.  80-83J  ANSWERS  39 

49.    6,2.  51.  None.                           53.    i^+^f, 

.0.   2^^  52.  OiV^.              54.   3, -2,  ±2^ZT. 

55.  1,  -1,  -'Y^,   i±J^- 

Exercise  81. 

1.  a;-l,  y=:2  ;  0?=^^  2/=-  !•  H-  ^=^,  2/=2  ;  a?=2,  ^=3. 

2.  .r=3,  y=—2  ;  a7=— 3,  2/=2.  12.  x—4:,  y=l  ;  aj=2,  2/=:3. 

3.  x=l,  y=i  ;  .T=4,  y=^\.  13.  07=7,  ?/=4  ;  a;=— 4,  y=—7. 

4.  ic=— 5,  2/=2  ;  a7=-10,  2/=-8.  14.  a?=l,  i/=5  ;  ir=-3,  2/=-3. 

5.  x=5,  y=7  ;  a7=-ff,  2/=-ff.  15.  x=^,  y=l  ;  ic=|,  2/=-|. 

6.  ir=2,  2/=l  ;  x=-l,  y--=-2.  16.  ir--=i,  2/=| ;  x=i,  y=l 

7.  £c=2,  y=^  ;  a^^-H,  2/=-2.  17.  a;=ll,  y=-8  ;  a;=8,  7/=-ll. 

8.  x=^,  y=2  ;  £P=2,  y=i.  18.  a=3,  6=1  ;  a=l,  6=3. 

9.  a;=-4,  y=-3  ;  ic=f,  y=—\^-.  19.  ^=4,  2i7=12  ;  t=—^^,  w=-\K 
10.  a?=6,  ?/=l  ;  a:=l,  ?/=6.  20.  m  =— 8,?i=— 3;  m=-V,  n=-V^. 

21.  A=5,  B=z4:;  A=z4,  ©=5. 

Exercise  82. 

1.  07=5,"  t/=3  ;  07=— 5,  y='S  ;  a;=5,  ?/=— 3  ;  x=—5,  y——S. 

2.  07=6,  ?/=l ;  a?=-6,  y=l  ;  .t=6,  y=-l  ;  a7=-6,  ?/=-l. 

3.  x=5,  y=2  ;  a7=— 5,  7/=2  ;  a;=5,  y——2  ;  a;=— 5,  y=—2. 

4.  a7=ii/2,  2/=il/2";  x=-U/2,  y=iV2;  x=n'^,  y=r-ii/2; 
x=-^y2,y=-^l/2. 

5.  07=2,  2/=3  ;  a;=2,  2^=-3  ;  £C=-2,  ?/=3  ;  07=-2,  y=-S. 

6.  07=1,  2/=5  ;  x=l,  y=-o  ;  07=-|,  2/=r) ;  07=— |,  ?/=-5. 

7.  07=6,  y=i  ;  0^=6,  2/=  — | ;  a-=— 6,  2/=| ;  o?=— 6,  o;=— f. 

8.  07=i,  y=i  ;  07=i,  2/=-i  ;  x=-i,  y=i  ;  0?=-^  y=-h 

9.  £c=6,  2/=9  ;  o;=6,  y-—d  ;  07=-6,  jf=9 ;  07=-6,  y=-9. 

10.  o;=4,  y=2  ;  a;=4,  2/=— 2  ;  a;=— 4,  y=2  ;  07=— 4,  y=—2. 

11.  a?=4,  2/=3  ;  a;=-4,  ?/=3 ;  a?=4,  y=-3  ;  a;=-4,  2/=-3. 

Exercise  83. 
1.  x=2,  y=l',  x=^2,  y=-l;  a;=||/2,  y=iy2;  a;=-fi/2;  2/=-|l/2l 


40  ANSWERS  [Ex.  83-85 

2.  x=0,  y=yl9  ;  x=0,  y=-yid ;  x=S,  y--^  ;  a?=:-3,  y=2. 

3.  x=i/%  y--0  ;  x=-y5,  y=0  ;  a7=5,  2/=-3  ;  x=-5,  y=S. 

4.  a;=3,  t/=4;  ic=-3,  2/=-4;  x=lV%  2/=|t/3;  a7rr-||/3,  ^/zzr— fi/i". 

5.  ic=2,  2/=4  ;  x--'Z,  y--^  ;  a;=i/2,  y=^\/2;  .t~-|/ 2, 2/=-3i/2. 

6.  a;=il,  2/=2;.rr=-l,  2/=-2  ;  a7=|/3;  2/=0  ;  a;=:-|/3,  ?/=0. 

7.  a?— 1, 2/=5  ;  a7=:  — 1,  2/=— 5 ;  £c=14,  2/=— 8  ;  0?=  — 14, 2/=8. 

8.  x—2,  y=5  ;  a;=— 2,  y=—5  ;  ar=4|/3,  ?/=  — 1/3  ;  ic=— 4i/3,  y=V'd. 

9.  ic=2,  2/==5  ;  a;=:— 2,  y——5.     (Defective  system). 

10.  07=3, 2/=5  ;  a?=:  — 3,  y=—5  ;  x=S,  y=—5  ;  x=—S,  y=5. 

11.  (j?=:4,  y=5  ;  x=—4:,  y=—^  ;  x=Si/'S,  y=l/'S  ;  0?=— 3y  3,  ?/=:  — ]/3. 


12.  07=5,  y=l  ;  a7=-5,  2/=-l  ;  x=iy  -10,  2/=-il/  -10  ; 

Exercise  84. 

1.  x=~l,  y=2  ;  07=2,  2/=-l.  2.  a7=-2,  ?/=5  ;  £r=:5,  y=-2. 

3.  07=6,  ?/=2  ;  07=— 2,  ?/=  — 6. 

4.  07=5,  2^=-l ;  07=-5,  y=l ;  07=-l,  ?/=5  ;  07=1,  ?/=-5. 
5.  07=5,  2/=— 3  ;  07=3,  2/=— 5.  6.  07=6,  y=z5  ;  07=5,  y=G: 
7    o^-l£l±i  ^-1^5-1.  ^_t/5-1  l/5+l. 

\-Vl          -l-j/s".             l-T/5    ^_l-|/5 
^-      2      '  ^-        2 '  ^ 2 '  ^ 2— 

8.  07=5,  ^=4  ;  07=4,  2/=5  ;  07=— 5+i/— 14,  y=— 5— l/— 14; 
07=— 5— |/— 14,  2/=— 5+l/— 14. 

9.  07=7,  ^=-1 ;  07=-7,  2/=l ;  a7=l,  2/=-7  ;  07=-l,  2/=7. 

10.  07=6,  y-^  ;  07=-9,  2/=-6. 

11.  a;=5, 2/=-7  ;  o;=-7,  2/=5.  12.  a;=4,  2/=5  ;  o;=5,  y^L 

Exercise  85. 

1.  a;=3,  2/=2  ;  o?=2,  2/=3.  2.  o;=0,  2/=-3  ;  aj=3, 2/=0. 

3.  a?=5, 2/=-2  ;  o;=2, 2/=-5. 


Ex.  85-86] 


ANSWERS 


41 


4.  x=l,  y=2  ;  x=2,  y=l ;  a;=:f +il/-55,  2/=^-il/-55  ; 

5.  irrz— 2,  2^=:8  ;  x=8,  y-—2. 

6.  37=4,  y=l  ;  a7=-l,  ^=-4  ;  a;=f+il/-79,  2/=-|+il/-79  ; 

7.  a;— 5,  2/=3  ;  a;=5,  1/=— 3  ;  0?=:— 5,  y=d  ;  a;=:— 5,  y=~d. 

8.  07=3,  2/=^3  ;  x=3,  y=d  ;  a?=— 3,  2/=— 3  ;  x=—d,  y=—S. 

9.  37=4,  y=2  ;  a7rr-2,  2/=-4.  10.  x=2,  y--S  ;  a7=3,  2/=2. 

11.  x=S,  y=l ;  a;=l,  y=S  ;  a7=2+5|/^,  2/=2-5i/^  a;=:2-5v/-l', 

12.  x=S,y=2;x--2,'y=-S. 

Exercise  86. 

1.  x=25,  y=9  ;  x=9,  ?/=25  ;  ir=-81+8v  ^97,  2/=-81-8]/^^97'; 

.T=-81-8l/^7,  2/:rr-81+8l/^I^. 

2.  x=l,  y=4: ;  a7=4,  ^=1  ;  x=-S-i  ^,  y=-S+y^; 

3.  x=4:,  y='d  ;  07=3,  y=4: ;  a?=— 3, 2/=— 4  ;  a7=— 4,  y=—d. 

4.  .T=i,  2/==i;    07=-^,?/=-^. 

5.  x=2,  2/=7  ;  ^7=7,  y=2  ;  a7=f ,  2/=!  .  ^^,-7^  2/=|. 

6.  37=2,  2/=l  ;  37=1,  2^=2  ;  0;=  — 1,  y=—2  ;  a7=:— 2,  y=—l ; 


V-i-yn. 


^--V-^-yn  -yz-i+yii 

^-  2  '       2^ 2 • 

7.  07=8,  2/=2  ;  a;=2,  y=8. 

8.  a;=3,  y=l  ;  07=-3,  y=  —  l  ;  07=1,  y=S  ;  a7=-l,  2/=-3. 

9.  x=-i,  2/^796.  10.  a;=27,  2/=8  ;  x=-8,  y=-27, 

11.  ^=6,  y=n  ;  a.=3, 2/^6  ;  a.=Z:lW57,       ^..ziW:^. 
^_-19-3v57        .._-19+3v^57 


42  ANSWERS  [Ex.  86-89 

12.  x=l,  y=—5  ;  a?=5,  y=-l ;  x=l,  y=4: ;  x=-i,  y=—l, 

13.  Impossible  system.  14.   x=W,  y=Q. 
15.   x—4:,  y=6  ;  x=5,  y=4:  ; 

^_-9-l/T6i      ^^-9+1/161.    .^._ -9+ 1/161      .,_-9-i/167. 


2'^  2'  2'^  2 

16.  a;=0,  ^=0;  x^i,  y-\  ;  a?-!,  2/=-2  ;  x=—\,  y=  —  h 

Exercise  88. 
1.   4  and  9.  2.    5  and  11  ;  5  and  — 11  ;  —5  and  11  ;  or  —5  and  —11. 

3.    3  and  5.  4.    6  and  2  ;  or  -2  and  -6. 

5.    10  and  6  ;   10  and  -6  ;  -8  and  0.        6.    2  and  8.        7.    49  and  9. 

8.  5  and  6  ;     -2  and  -1  ;  or  2|/^  and  -2]/^. 

9.  1  and  4 ;  1  and  —4  ;  —1  and  4  ;  or  —1  and  —4. 

10.  62.  11.  36.  12.  10  and  4. 

13.  4  in.  and  9  in.;  or  6  in.  and  6  in. 

14.  2  in.  to  width  and  1  in.  from  length  ;  or  3  in.  to  width  and  2  in. 

from  length.         15.   6  and  8.  16.   $96. 

17.  Derrick  40  ft.;   guy-rope  50  ft.     18.    40  feet  and  25  ft. 

19.    f .  20.   $3.25  ;  10  days.         21.    6  men  ;  8  days. 

22.  $32  for  apples  ;  $22  for  potatoes. 
23.   $250  ;  6%.  24.   8  days,  and  12  days. 

Exercise  89. 

7.  a?>5.  9.  a!<6.  11.  a?>3or  <~1. 

8.  x^\K  10.  ic>ll.  12.  a;>5or  <-3. 

13.  X  between  Y-  and  7. 

14.  X  between  a  and  -,  when  a  is  not  1. 

a 

15.  X  between  2  and  6.  18.  Impossible  system. 

16.  X  between  —14  and  —3.  19.  x  between  4  and  12. 

17.  x<C-4.  20.  X  between  -IfJ  and  1. 

21.  Values  of  a;>5  and  ^<5  that  satisfy  equation. 

22.  Values  of  a7>3  and  y<^5  that  satisfy  equation. 

23.  Values  of  a;<l  and  2/>— H  that  satisfy  equation. 


Ex.  89-92]  ANSWERS  43 

24.  Values  of  a;^  — |f  and  y^rii  t^**  satisfy  equation. 

25.  Values  of  ic^-— ^  and  y^:^r\y  that  satisfy  equation. 


1. 

-V-. 

2. 
3. 

f. 

1 

3i- 

4. 

h 
a* 

5. 

._-_, 

6. 

T^. 

7. 

11,  li  f. 

8. 

/.. 

9. 

|. 

10. 

3 

2(a-5)' 

11. 

4. 

52.  320  yds  ;  4* 

54. 

|.                    55. 

1. 

x=^y. 

2. 

42. 

9. 

V=:32t. 

12. 

16  cu.  in. 

13. 

256  ft.,  113  ft. 

Exercise  90. 

12. 

4. 

24. 

pq\ 

13. 

15  and  20. 

25. 

tV. 

14. 

15  and  21. 

26. 

^ 

15. 

4  and  18. 

a ' 

16. 

48. 

27. 

X  ' 

17, 

150^. 

28. 

4. 

18. 

i'2. 

29. 

45. 

19. 

TO. 

30. 

3iu*. 

20. 

he 
a' 

31. 

3v^6. 

21. 

y_z 

32. 

Vab. 

X 

33. 

Vxy. 

22. 

4. 

34. 

2ad 

23. 

9a;6. 

"3c"' 

is;  480  yds.  53.  3  or  4. 

55.    8  and  12  ;  or  -12  and  -8.  56.  64  ft. 

Exercise    91. 

3.  xy=24:.  5.   x=8yz.  7.   7. 

4.  25.                       6.    2.  8.   x^=4y^. 
10.    30sq.in.                      11.  201.0624  sq.  ft. 

14.  2.44  approximately. 

15.  7^^^  ft.  from  end  of  heavier. 

Exercise   92.  ' 

1.  F—km,  where  -F=  force,  ?7i=mass,  and  ifcrra  constant. 

2.  a—     ^2    ,  wherea=attraction, d=distance  between  tliem,  m=rmass 

of  one,  m'=:mass  of  other,  and  k=?L  constant. 

3.  t= — j — ,   where  f=tension,  m^mass^  v=velocity,  ?=length,  and 

fc— a  constant. 


44  ANSWERS  [Ex.  92-94 

4.  t=1c\/ 1,   where  t=time,  l=length,  k=£L  constant. 

5.  p—khh,   where  p=pressure,   7i=depth,   &=:area  of  bottom,   ^-—a 

constant. 

6.  H8^=k,  where  H=hent,  s— distance,  A;=a  constant. 

7.  R=k-,    where  72= resistance,  Z=: length,    a = cross-sectional  area. 


a 
and  k=R  constant. 


Exercise  93. 


"•    ^-  12  ^  20.    A- 

_  "■  ac2-  «+b  28.   8i/a. 

A  3—  13.  -o^-  "  — ^  1 

5.    ^gi,  14.  M.  '^-   SMi^)-  ^l/^ 

V.    ^x2.  ~       24.   -^• 


16.   |«V.  a;")/^  «.- 

52  31     y  if. 

17     '^^?/'^  25. 1-  •    ,V- 


9.   y  27.  1                                  0 

10.  |/(|7.  IS-  56^-          ^^;  "^4           32.  ^>(^:^2 

33     V^^+b  .          34.  27.                                 36.  i. 

l/a^'  35.  4.                                   37.  ^k- 

38.  |.         39.  V.         40.  ^i^.        41.  ^j%^,         42.  /^.         43.  |.          44.  24. 

Exercise  94. 

1.  2a;2.  6.  a^, 

2.  1. 


10.  i/m\ 


3.  a.  '•   ^ -•  12.  ab. 

4.  4^. 


61/c 

&  ^  15b 

5.  -5-^.  9.  -^  14.  -rz::. 

l/a8  V  a46**  l/aii 


Ex.  94-95] 

ANSWERS 

45 

15.   ?5. 

a 

le.  a46i2. 
17.   ^^^j^^ 

22. 
23. 
24. 
25. 

a2aj. 
xyVx^^yK 

30. 

31. 
32. 

3 

X+y 

V^-y' 

— s. 

18.  la'y'x. 

X 

26. 
27. 
28. 

1 

33. 

34. 
35. 

a7 

a;3+a;^2/^+2/«.                      ' 
a2-l.                                     , 

^ifiy^ 
20.  a. 

xyi/xy^' 

21.  a. 

29. 

-1. 

36. 

x-y.                                     j 

37.  8a^2_i8ar-i- 

38.  a2-62. 

39.  x+y. 

47- 

-1507. 

44. 
45. 

xi- 

+  1965+36-115-5-6&-10.                 i 

40.   ^+27. 

46. 

x-^ 

^+2a;-* 

+3+2a!^+a;*. 

41.  2a-20a^+18a~*. 

47. 

a4- 

-1+a-* 

42.  a;^— £c32/^+2/^ 

48. 

aJ+2aW 

^6-1. 

43.  a;"^+a;~^+a7' 

-^+1. 

49. 

a^i 

—2a?  V+y- 

50.  X- 

i+2a;-^+3aj~^+4a;"^+5+4a^+  ^x' 

^+2a;*+a?. 

51.  a^-a?~i 

52.  9a;-' -162/5. 

56. 
57. 

a-3+a-2+a-i+l.                                   1 

53.  -\/a+2i/h. 

58. 

x^+x^y^+x^y^+y^.                             \ 

54.  ar-2-3ar-i+9. 

59. 

h 

' 

55.  a-»+3. 

60. 

18. 

V 

Exercise  95. 

V 

■^v,,^^^^ 

1.  24. 

6. 

336. 

11. 

30. 

16.  240.    ^                         1 

2.  720. 

7. 

60. 

12. 

180. 

17.  12. 

3.  720. 

8. 

TliuU 

13. 

24. 

18.  56. 

4.  48. 

9. 

A. 

14. 

1814400. 

19.  56. 

5.  7. 

10. 

8|. 

15. 

380. 

20.  16380. 

40  ANSWERS  [Ex.  95-1 

24.  do. 

25.  16. 


ANSWERS 

[E 

26.  650. 

28.  56. 

30. 

!l£: 

27.  120. 

29.  103^800. 
32.  6 ;  10080. 

Exercise  96. 

31. 

90. 

6.  56. 

11.  66i,2^.0  i 

;22a. 

16. 

56. 

7.  9. 

12.  10. 

17. 

270. 

8.  1140. 

13.  10. 

18. 

28.     ' 

9.  53130. 

14.  325. 

i 

19. 

560. 

10.  2. 

15.  4. 

20. 

53130. 

21.  21. 

•       22.  45. 

Exercise'  97. 

1.  4. 

2.  210. 

3.  82160. 

4.  455. 

5.  220. 


1.  oc^+5x^y+10x^^+10ai^+5xy^-]-y^, 

2.  64ic«-576a76y-i-2160a-V~4320a^+4860a;V-2916a;^+T292/«. 
"'    H-8a;2+24.r*+32if«4-16a^. 

•:    l28ai*-h448ai2&-l-672ai062+560a823»»+280a6&*+84a4664.i4a266+57, 
1  ~  15a«+ 90a«-270a»+405ai2~  343«i5. 
x'^-6aTlOa24-  15a^a*_20ic«a«4-  15a?*a8  -6a'2«io-+-  a^^. 
1  -f  16.T+112a?2+448a^+1120a^+1792a--5+1792cc«+1024a77+256a^. 

ar-*+4ar-2+6+4r2-fa?*. 

32^10  ^_  80ar-«-+-S0;r-«+ 40ar-4+ lOar-2-i- 1 . 

a«- 6a65^+15a4b- 20a«6^+ 15a262- 6aZ)^+fe8. 

,.      5a^h    5am    20a2?)8    40a6* .   326* 
"^-    32^24*^    T"+    27~~+~8r"^"2ir' 

•      r      f-9a    W+ 36a    ^b^+8ia     64-12('.    '^^ "  ^'^  ■   ~  '  '     - 
♦-36a"'^&^+9«''V^+68. 

'.  18.    -8064a;iV-  19-    -36a?-ia   '. 


Ex.  97-98J  ANSWERS  ^^ 

20     ^*^^?^  oo        ^•'>''>04 

81     '  "*    — — isT"" 

21.    ^%%ayc-^.  24     _,^^.22 

22-    210a;i  25.    l+dx-5o^+Safi-afi. 

27.    16-82a+24a2-8a8+a4+3262-48a62+24a262-4a862-f.'M64-24a6* 
+6a264+86«-4a&6+58. 

>?8.    l+3a!+6a;2+4ic3-6a^-2ic6+3a*-.'r9. 

29.  «3-53+c8-#-3a26+3a52+3a2c-3c2d+3cd2+362c-3a2d!-3t2d+3ae2 

+3ad2_36c2-35d2_6a6c+6a&d-6acdH-66cd. 

30.  1 6a4-  32a864-  24a262-8a68+ 64+ 96a8c2- 1 44a26c2+  TZab^c^-  12b»c^ 

+216a2c*-216a6c*+5462c4+216ac«-1086c6+81c8. 

Exercise   98. 

1.  l+2a7+3a;2+4a^+ 

2.  i-ix^+ix^-j\afi+ -' 

3.  a-8+4a-i0a;2+l0a-i2a^+20«-Wa^+ , 


"*•  32a:^  ^64a^^  128.x' '^25()a,-8'^ 

5.  l-^x^+Gx*-10j(fi+  •-  •  . 

6.  l-lx^-lx*-^\afi- 

7.  a*+fa"^62_^^^-f54_^^.5^^-l56. 

8.  ^+-J_+       3,5 


l/2£P    4i/2£cs    32i/2ic6     1281/ 2a-7 

9.    v^-iv^a;+iv^2if2-j|v  2arV 

1.0.   a;~*-|a;~^+||a;~~'^*_ma;~'oV • 

11.  i/3+ — 7^- -+ -(-)'  '  '  • 

^    '^2]/3    241/3    1441/ 3^     ^ 

12.  i%>---37:~^ — H7:- 5--- 

^  ~     3i/4     18|/'4    3241/4 

13.  ^V?^'-  .  15.  5e-5. 

14.  5ligf||T«~'^'ci  16.   — s^Vj-^i^". 


48 


ANSWERS 


[Ex.  99-102 


1. 

2. 

3. 

4. 

5. 
21. 
26. 

1. 
2. 

10. 
11. 
12. 
13. 
18. 
19. 


1.96794-.. 
2.9925+. 


146. 

37. 

20i. 

329. 

3. 

7500. 
10,  6,  2... 

1679616. 
-16384. 


Exercise  09. 

3.  6.0276+.  5.  4.0193+.  7.  4.9984+. 

4.  2.0800+.  6.  1.9947+.  8.  5.0990+. 
3.0006+ .                                 10.  2.7589+. 

Exercise  100. 


6.  3^. 

7.  49 ;  81. 

8.  14i ;  26i. 

9.  16th. 
10.  11th. 

24.  9,  13,  17,  21. 
27,  $28.50. 


11.  12. 

12.  270. 

13.  416. 

14.  2500. 
16.  2550. 


16.  97i. 

17.  5  or  6. 

18.  64. 

19.  2475. 

20.  19800. 


25.'  -8h  -7,  -H,  -4,  -2i,  -1. 


28.  at^ 


Exercise  101. 

5-  jAi- 


3.  34U. 


^'  -^-  6.    ±12i/2. 

9.  T25+5V+i-l-i+2+8+32+128. 


29.  8. 

7.  -8^. 

8.  i. 


-50,  20. 

1/2,  2,  2i/2. 

24. 

Iflll^.  20.  4. 

3.  21.  22i. 

26.  60. 

28.  |+|+^«^  +  3-V5+ 
36-12+4-1+ 

A%'  34.  IH. 

I  "35.  I 


14.  Hi 

15.  -2730. 

16.  971.2+, 

17.  42. 


24.  1. 

25.  3i. 


27.  -j\. 

or  ^+l+^%+jh+  '  ' 
31.  A 


30.  |.' 

37.  56^«^  ft. 

38.  4  years. 


39.  $1.21i. 

40.  149||ft. 


Exercise    102. 
1.  A.  2.  ;,V  3.  ^%.  4.  ^V 

6.  -i'^-HKA  +  tV+3\  +  iV+-.      6.  0+4+2+|+l  +  t+  • 


Ex.  102-105]  ANSWERS  49 

7.  i.  8.  3.  9.  -2,  -6,  6,  2. 

10.  hh  ^%  1%,  ^»  if.  11-  80+41/899,  80-4V/399. 

Exercise  103. 

1.  l+x+x^+x^+oc^+  '  '  '  .  2.  l+2a:+2a;2+2ir8+2ii;4+ , 

3.  2x—Bx^+dx^-dx^+Sx^- 

4.  l+x'^+x^+oc^+x^+ 5.  x+x^+oc^+oc^+x^+  .  .  .  . , 

6.  i+^x-^x^+j%x^-^%x^+ 

7.  l-x+Sx^-5x^+7x*- 

8.  2x+5x^+Ux»+S'7x*+97x^+  •  •  •  . 

9.  x-2+2x-^+2+2x+2x'^+ 

10.  ocr-^—Qc—^—x-'^—l—x—x^— 

11.  x-^-h4x-2+17x-^+7S+S09x+  ■  •  •  •  . 

12.  x^-x^-2x^+7x^-Sx^- 

13.  x-^—l+x—dx^+5x^- 

14.  |aj-2-fa^i+i/-ifa;+|fic2_  •  •  .  .  . 

15.  2xi^-2x^+5x*-7x^+12x^- 

Exercise  104. 

1.  l-ix-ix^-j^s^-j^^x^- 

2.  l+|a7—V-i»2+W^--rlF^+  •  •  •  • 

3.  l-2£C-2ic2-4a^-10ii^-  •••..•. 

4.  l+^x-^x'^+j\x^-j%%x^+ 

5.  l-j^x+^x2+j%x^+j^^x^- 

6.  2+ix^—,\x^+^\-^a^-j^^^^x^+ . 

7.  1—x—ix^-ix^-^x*-  •  •  •  •  . 

8.  l+x~x^+x^—^x^+ 

9   2i/34.j/L8_l/2^  .    1^3  9__5i/2    ,2  ,    .  .  . 

10.  l+x+x-\  11.  l+2a;+3a;2.  12.  4x^+3x-5. 

13.  4.-dx+x^-2x\  14.  l-2a+4a2-8a8. 

Exercise  105. 

1.  x=y+y^+y^+y^+ .  2.  x=y-2y^+5y^-Uy*+ 

3.  x=y-iy2+ly»-^\y^+ 


50  ANSWERS  [Ex.  105-107 

4.  a;r=2/-32/2+13^3-672/4+ 

5.  x=y+2y^+4y^+8y'^+ 

6.  07=3?/+ V2/2+ W/2/^+-¥tV2/'+ . 

7.  x=y—y^+2y^—ny^+ 

8.  x=y+ly^+j%y^+^\\7f+ 

Exercise  106. 

1  1    _    1  12  1 ?_ 

•  1-x    T+x'  (a?-l)3    x-l' 

2  -J_+_A_  13.    l--i- 

3  8       _       1 5__  14     3__2 8 

3(a?-2)     6(07+1)     2{x-iy  '   x    x+l     (x+l)'^' 

4  7  8  _    1     __      1  +  3^ 

•  3a;-2^2ii;+3'  2^-3     9+6^'+4i^'2- 

5  _J_  +  _:L__1_.  16        ^  1     I       -^'-1 

'    a?-l     a?-2    £c+3  '   ic-1     a'4-1     (ic^— ar+l)* 

A.         ^5  35 10_  17     A-1+I-_J_+       3 

4(2a?+l)^12(2a;-l)     3(a;+l)'  ar    a;2^a?3    a;-l^(x-l;2' 

^'  x+l    a?+2    a;+3'  2{x-^+x+l)     2(x^-x+iy 

8    _J x-2  19    _J 2  2 

3(07+1)     3(.T2-a;-+-l)'  '   07-l     a?+l     072+0;+ 1* 

2  12  1  20       2a?-7    _2x'^+2x-4: 

^-   207-3"^  (207-3)2     {2x-d)^'  '    3(x2+2)       3(0^3+2)    ' 

ID       1     I        2    , 1_  21  16  16  4 

o;-!"^ (07-1)2    a;-2*  '    {x+2y^     (x+2)^    x+2' 

11  _i __i i_      22     3-42 

•  4(07-1)     4(07+1)     2(^24.1)-  •    (i_a;)3     (i_a;)2^i_aj- 

23       1.1,         1  1 

07+1      0;— 1      a;2+a7+l      072— 0?+l' 

24.    l-i  +  — 1 _L_ 

078    07^(a;+2)3    o;+2' 

Exercise  107. 

1.  log232=5.                      4.  Iogiol0000=4.  7.  logg^V^-^ 

2.  log381=4.                       5.  log5l5625=6.  8.  logiojU  =  -'^' 

3.  log7343=3.                     6.  log4^\  =  -3.  9.  32^9. 


Ex.  107-109J 

ANSWERS 

10.  2*=  16. 

16. 

3. 

22.   -2. 

28. 

4. 

11.  42=16. 

17. 
18. 
19. 

3. 
2. 
-4. 

23.  2. 

24.  3. 

25.  -3. 

29. 

64. 

12.    8^=4. 

^0. 

3) 

13.  7-2-^V 

31. 

!».. 

14.  10-3=.001. 

20. 

3. 

26.  0. 

32. 

h 

15.    100~^=.001. 

21. 

-1. 

34. 

27.  0. 

33. 

1034. 

51 


Exercise  108. 

1.  logaOC+\ogay+\ogaZ+\ogaU^'-  3.    log„2+  ]ogaX  +  2  logaV  +  S  loga 

2.  2  \oga00+2  logaV.  4.    2  logaX+logay-^  logaZ. 

5.  i  logaX+^  \0gay-l0gaZ-i  logatV. 

6.  i  \0gaX+t  loga?/— i  logaZ—^  logaW. 

7.  logaX-logay-logaZ-logaW. 

8.  l0gaa;+2  logay-^OgaZ-2  ]ogaW. 

9.    2(l0gaa7  +  l0ga2/-l0ga;^-l0gaU').       H-    i  \ogaX+^  ]ogay-i  logaZ. 
10.    I  ]0gaa?-f  logay.  12.    i  I0ga^+l0ga2/  +  i  log„a?. 

13.  j  logaX-i  logay +i  logaZ-i  \ogaW. 

14.  l0ga2  +  logaa;+2  loga?/-loga3— loga^:-^  logaW. 


15.    log„^^. 

21. 

>og»| 

30. 
31. 

2.3222. 
3. 

^«-    ^^^«^- 

22. 

1.2552. 

32. 

-.8239. 

23. 

1.1761. 

n.  iog«,^:. 

24. 

1.3010. 

33. 

.6276. 

18.    logj/--. 
1/2/ 

25. 
26. 

1.6990. 
1.3801. 

34. 
35. 

.7385. 
.7517. 

19.    )logal/  ff^  ' 
^loga        "^ 

^^. 

27. 

.3680. 

36. 

.2762. 

28. 

.2219. 

37. 

-.3460. 

Vy+z 

29. 

1.8451. 

38. 

-.0537. 

Exercise  ! 

109. 

1.  2.3324. 

4. 

2.8555. 

7. 

2.9191. 

10.  .9294. 

2.  2.8280. 

5. 

.6646. 

8. 

1.2041. 

11.  2. 

3.  2.9731. 

*      6. 

1.2989. 

9. 

1.4624. 

12.  2.9542. 

52 

ANSWERS 

[Ex.  109-113 

13. 

0. 

17. 

2.0021. 

21.  2.8353. 

25. 

1.99996. 

14. 

.3010. 

18. 

2.9670. 

22.  3.6302. 

26. 

3.3079. 

15. 

3.2263. 

19. 

1.9361. 

23.  .0398. 

27. 

3.9208. 

16. 

1.5349. 

20. 
29. 

3^4983. 
4.8363. 

24.  4.0792. 

30.  2.8739. 

28. 

r.7025. 

Exercise  110. 

1. 

57.3. 

6. 

488000. 

11.    10223.8. 

16. 

105195.12. 

2. 

7270. 

7. 

301. 

12.    29.53. 

17. 

1094.5. 

3. 

1.53. 

8. 

4000. 

13.    .09474. 

18. 

.0002989. 

4. 

.289. 

9. 

.00531. 

14.    701.5. 

19. 

7.4733. 

5. 

.0427. 

10. 

790. 

15.    .2554. 

20. 

37.445. 

Exercise  111. 

1. 

8.1379- 

-10. 

5. 

8.9087- 

-10. 

9.   5.7368-10. 

13. 

1.5017. 

2. 

7.1605- 

-10. 

6. 

8.1658- 

-10. 

10.    5.3649-10. 

14. 

4.2055. 

3. 

9.3410- 

-10. 

7. 

6.8617- 

-10. 

11.    2.4935. 

15. 

3.3260. 

4. 

7.4859- 

-10. 

8. 

7.5829- 

-10. 

12.    .1645. 

16. 

3.2489. 

Exercise  112. 

1. 

23.41. 

4. 

5428.6. 

7.    .001228. 

10. 

.8224. 

2. 

.004165 

5. 

-.08936. 

8.    103.58. 

11. 

.0564. 

B.    150.9.  9.    -526.05.  12.   4.947. 

13.    1817500.  14.    .000000000002,  approx. 

15.    14.81. 

18.  4.295. 

19.  -4.1166. 


16.    .004541.  17.    1.561,  approx. 

20.  .001204.  22.    .04327.  24.    5.1811. 

21.  .387.  23.   3025.7. 


25.    .00000285. 


26.    14.53. 

27.    85.6,  approx. 
Exercise    113. 

1. 

4. 

4.   2.4651.                7.   2.165.  ^ 

10.    13. 

2. 

2. 

5.    1.5481.                8.   3.3852. 

11.    5. 

3. 

3. 

6.   8.6336.                9.   2.998. 

12.    1000. 

13. 

10000. 

14.   2.209.         15.    1.631.            16.    1. 

17.   i. 

ANSWERS  TO  REVIEW  EXERCISES 


Exercises  for  Review  (I). 
5.  16|.  21.  45  ft.  and  15  ft.         27.  -$500. 

8.  8,  16,  32.  22.  105  and  21.  28.  +50«  and  -15°. 

11.  (a)  13f  ;  (&)  230f.        25.  +75  and  -15.  30.   -3. 

12.  (a)  46;  (5)9.  26.  -15.     60  lbs.  32.  32. 

34.  64 ;  -27  ;  ^V  ;  tV-  36.   -4  ;  4  ;  -27  ;  |. 

Exercises  for  Review  (II). 

1.  -y^-xhj.  .  5.  lla;2-3a;+2. 

2.  2ab-3a;-2c+5a;2.  7.  a^+2a'^y +2x^+0^-^1/^, 

3.  9a;2-92/+2a.  8.  2«8-6a2+a-5. 

4.  (3?/2+2a4-3cy)a?.  11.  2a-4&-2c. 

13.  (7-a)a.^+(7-a5).T8+(5-3c2)a;. 

14.  -  {h-^+c)o(fi-  (a+26-2)a;2-  (4a-b)a;. 

15.  a2c^+{a-^)x^+{h-2)x+a^+h-^2. 

16.  a?;  a";  x^K  17.  -3a8aV- 
19.  2x'5— 4a^+6.T3-10a;2. 

21.  _3a*+5a46-5a362+4a268-2a6*+266. 

22.  40a2_35a5.  23.  a^ ;  a;  a^ .  ^8, 
25.  Quotient-a4+2a364-3a262^-4ab3-2a66+55*— 467 . 

Remainder=6a5S-6a68— 466+469. 
29.  |a;4^.  30.  a^  ;  a""* ;  b'^'m^" ;  S^. 

31.  £c5//5  ;  £ci22/i8 ;  64a96a>. 

32.  -8a^2/i%i2  .  81a2464ci2ci8 .  -^^. 

33.  a2+2a6-t-62;  Ax^-\2x^j^+^if  ;  ^a*+2a26+962 ;  0^-4^2+4. 

34.  rt2_|_52_,_c2_|_2a5+2cic+26c;  Ax'^-\-^y'^-\-z^—\2ocy+ixz—Qyz. 

53 


64  ANSWERS  [Ex.  for  Rev.  II-III 

36.    ±2a^;  Sa% ;  -2a^b^ ;  {a—hY\  imaginary. 

38.  Ax'^-a  or  a-4x2  ;  9+5n3  or  -9-5n3  ;  2x'^-^y\ 

39.  a2-b2;  l6a^_9?/4;  ^-86076. 

40.  x'^+{a+h)x+ah;  a^+2a;2-24;  4a2624.20a6+21 ;  i^c'^x'^+^^cx-^Z. 

41.  ic2_f.a:^^2/2 ;  £t7^+£c2^2_|_|^4  .  a:^_a:22/2_|_2/4  ;  a^—a^+a^—1. 

42.  When  ?i  is  even.  43.  When  n  is  odd. 

Exercises  for  Review  (III). 

3.  (a)  2x{Zx^-2xhj+'f);  (b)  a^b(da-2h+l);  (c)  2x'^(x^-3); 

(d)  ab{a-b){a^+b^)',  (e)  x-y«-Hx^+y);  (/)  a-b-ia^+b'). 

4.  (a)   {xy^-l)(xy^+l)',  (6)  ia;(a?-62/)  (a!+6?/); 

(c)   (a-6+2c3)(a-5-2c3);  (d)   (a+26+c)  (a+26-c); 

(e)  (ir+2)(ic-2) (0^2+4).  (^)   (l_£c2+22/2)(l+a;2-22/2). 

5.  (a)   (a:4-2)(ir-2)(x2-2£c+4)(a;2+2a;+4);  (6)   (a2+9)(a4-9a2+81); 

(c)  (l+ar2^2+a^)(l-a;2|/2+a^);       . 

(d)  (x-l-y)(x^-2x+l+xy—y+y^). 

6.  (a)   (a;-6)(a;+5);  (6)   (Sx-2)(x+l);  (c)   (3+4a:)(l-2ir); 
(d)    (5a?2-2/)(a^2+22/);  (e)   (2a:+32/)2;  (/)   (5-a^)(3+a;); 
(r/)   (a4+a2+l)(a4-a2+l);  (/i)   (6-2a)  (&+4a). 

7.  (a)   (a+6)(a+l)(a-l);  (6)   (£c-a)(a;+l)  (ct2-£c+1); 
(c)   (a?— 3+a— b)(a'— 3— a+6);  (d)   (x+l)(a+b)(a+c). 

8.  (rt)   (a;-l)(a;+2)(£t?-3);  (6)   (.t+2)  (2a'-l)(a;+4); 

(c)  (a-3)(a-3)(a-5);  (d)   (6+3)  (d^.,, 4). 

9.  (a)  2aa;(2aa;+3a2^-2/);  (6)  5a3a^(a+a;)2  ;   (c)  (2a:-y)2; 

(d)  (a;i/+16)2;  (e)   (2a;2-1.5mn)2; 

(/)  (2a7- 152/2) (207+151/2);  (l-14ir2/=^)(l  +  14cc2/2);  (gr)  (x^+y2)(x+y); 

(h)   {x+2)(x-y);  (i)   (a-b){a^+b^);  (j)   {dn-4m)(2-7m'^); 

(k)  {x-y+2)(x-y-2);  (I)   (a-b+5)(a-5-5); 

(m)   (d+3c+l)(d+3c-l);  (n)   (5.T+72/2f)2 ;  (o)   (iri/3+3)(a:2/3+9); 

(1))   (a72/-llz)(aj?/-13^);  (g)   (x-b)(x-24)', 

(r)   (itV+i2)(a-22,3._io);  (s)   (ir2+i)(a;+l);  {t)   (a-b)(a^+b^); 

(u)   {a+b+2ab){a+b-2ab);  (v)   {x^-\-y^){a+b){,a-b); 

{w)   (2a;- 3) (07+4);  {x)   (a+b)(c-d). 


Ex.  FOR  Rev.  III-IV]  ANSWERS  55 

10.  (a)   (3a8+l)2;(6)  (4a+56)(3a-26);  (c)  (a+5)(a-b)(a+25)(a-2?>); 
(d)   {2x^-4xy+dy^)(2x^+ixy+2y^);  (e)   (8a+96)(2a-6); 

(/)   (a-i-h+c-d)(a+h-c+d);  {g)   (x-+,l)^;  (h)   (a+h){a-h-l)', 
(i)   {x^+xy+y^)(x^-xy+y^);  (j)   (Sx-2y)(2x-Sy); 
(k)   {7x-Sy)ix+4ty)',  (l)   (4a-7)(4a-5);  (m)   (a2'»-?>'»)(a2'»-hi!>"); 
(n)   (fl+6)(a2-6a+36;;  (o)   (x^+2xy+y^+l)(x+y+l)(x+y-l); 
(p)   (a2+a+l)(a2-a+l);   (g)  3a;2^(a;2+(CT/+2/2)(a;2-a;2/+i/2); 
(r)  a(a-l-a5-5);  (s)  (l+rt)(l-b);  (f)  (a+5+c)(a+6-c) 

(c+«— b)(c— a+6);  (w)  {2a+b){dc—2d)',  (v)  {ax+hy){bx+ay); 
(M^)  (a+cc)(a-a;)(b2+a:2);  (a?)   (3a-l)(2a+36). 

11.  (a)  aj-2;  (6)  a?— 5  ;  (c)  2ic-l ;  (d)  £t'+3. 

12.  (a)  6a;2(i»-3)2;  (5)  10a26(a-5)(a+l);  (c)  a?2(a;-3)2(a;+2); 

(d)   {dx+5){2x-l)(x^+^). 
iPi    /^\         4ic2-19a;+24        .  /m       18-6^_  .  /^x      22a?+42 


art-6x'3+7a;2+6a;-8  '  '  '  a?2-2ic-48 '  '  '  a?2_,_i0ic+21* 
26-2a 

18.    (a)x+7;  {b)dx+Q;  (c)  3a?+30. 
19.    rJr-::,.  20.    €  21.    a.  -         1- 


x2^yr  «"•   a;2  •  ""'    2ic2_i' 

no     /   XA     /^,^^     ,  .  a^+h     ,,.a^—Qax+x^      .  .  x^z+x—y^z—z     ^.. 
23.    (a)  0  ;  (6)  1  ;  (c)  ^:p^,  ;  (d)     ^,^_^^,     ;   (e) ;^3|^—  ;  (/)  1. 

Exercises  for  Review   (IV). 

5.  (a)  ^',  (6)4,  (c)i. 

^     ,   ,        1+x  1  ...  2ci       .     a+2d     ,    2at—a 

6.  (a)  a=-2^,   a;^^^^^  ;  (5)  a=^^-^,   i^-^ST^   ^=—2"  ' 

...         .  i  *      J      *      J      a         d     .    d 

(c)  t=pr^,  i)=-^,   r=-^,    ^=-  ;  d=vt,  v=-^,   t=-. 

7-2  8    —  9    ^^ 

^-       "•  ^-    bx  ^'      2    ' 

,n     f—M-      r,-f^       n-J^  11     7«-:ST     „-V^     r-— 

12.    1.  18.   (a)x=S,y=-4;  {b)x=hy=^. 

19.   (r=— 1,  2/:=a+6;a=^,   6=r^. 


56  ANSWERS  Ex.  for  Rev.  IV-V] 

20.   x=z4,  y='S.  21.   x=a-\-h,  y—a—b. 

Exercises  for  Review    (V). 

6.  (l-6a2+126)x?^3a. 

7.  (aic2?/+aaj22;+3a2a;+ 12?/+ 12;2;)  1/^+7. 


8.    61/ 7.  13.    ^/ 635000. 


8/ 6,——       6/-- 


10.  1/125,  1/121,  l7l3.  15.    3j;i/243£rii. 

11.  i^8a^  16.  les+isiy'Io+ssi^m 

12.  v^fO.  17.    2-1/5" 

19.  f{+|fi/3.  20.  Z2aWi/ab'',  50a^i^3aj;  2yy'3x^y. 


21.  i/3a;22/3;  i7a;+3  ;  dx^\/x-y. 

22.  (a)  5+1/6  ;    (6)  15+4i/l5;    (c)  111/?  ;    (fi)  x'ul/loT 

27 


(e)  i/a?s^io  ;    (/)  Aa^  ;    (gr)   37-2+^   ;  (h)i/2. 
2b'.   2ic3|/Zi.  27.    (a)  8ai/^;  (&)  -24^^.  ;  (c)  3  ;  (d)  2-i/5. 

28.    3-1/^.  30.   -8+21/15. 

31.  M-^Si/35+Ai/^+At/^. 

32.  -^1/3+1^:^-1+11/31/-^. 
36.   (a)    ±il/3;  (b)    ±3;  (c)    ±  l/3. 

38.  (a)  i  or  -5  ;  (b)  |  or  -|  ;  (c)  i  or  -|. 

39.  (a)  8  or  -4  ;  (5)  i  or  -5  ;  (c)  a  or  |. 

40.  (a)  i  or  -f ;  (b)  i±il/l3  ;  (c)  5  or  | ;  (cl)  f  or  -5. 

41.  Cor  3.  42.  13  or -J/.  43.  l±v^^^^.  47.  ^. 

6 

49.  7a;2-19ar-6=0  :  x'^-2x-ji=0. 

52.   (a)  1/3,  -1/3,  1/2,  -1/2 ;  (5)  -1,  -1,  ^±|i/IT; 
(c)  1,  1,  -3±2i/2. 


Ex.  FOR  Rev.  V-VII]  ANSWERS  57 

58.  (a)  i+ll/19;  (6)  |;  (c)  0. 

59.  Cube  roots  cf  1:  1,  -i±il/^;cube  roots  of  8:  2,  -lij/^  ; 
cube  roots  of  27:  3,  -f  ±fl/^. 

63.  x-1,  y=^2  ;  x=2,  y-\  ;  £»7=-3+i/"^,  2/^-3-]/^  ; 
.T=-3-i/^,  ^=-3+1/ -2^ 

64.  A-TTi'^  65.   F=|7rr3;  Fi^^ttDS. 

Exercises  for  Review  (VI). 
6.  No.  22.  3. 

9.  4£n2  ;  i2cc^y2 .  6(a+6)2  ;  |/^.         37.  «i5  ;  «*  .  ^,15  ;  a'^n  .  «^. 

10.  64;  256«4;  8?/3 ;   (a+6)5.  '         '       '         '    ^^     '  ^^'* 

11.  t  ;  2.T22/ ;  GOZ>a; ;  a^+2/.  ^9.  a^ ;  0-2  ;  64aV?^- 

17.  |.  30.  8;  4;  5;  ^ 

18.  .^=10^.  „„     b2  .  2a253  ^       2 


19.  60^=2/.  •   a2  '    31/     '  (a-6)2' 

21.  1.  33.  i;  5;3V;i;  -32. 

3  —        4    —  ?/^  ^^ 

34.  .aio  ;  i/a2  ;  Va^  ;  ^/-,  ;  g^^' 

35.  £c"3+2a;~^2/-i+?/-2  ;  4a-2-12a-i?>~^+95-i  ; 

x-'^+^x'^y^ ^^x'^y^ +y  ; a;"^-2/~^  ;  a/^+h^  ;  a^+a^b^+b^. 

38.  a  greater  than  2. 

39.  a?  greater  than  V,  2/  greater  than  ^-f. 

40.  19.  41.  Between  8  and  15. 

Exercises  for  Review  (VII). 

3.  n{n-\)  ;  60;  5040;  ?i(n-l)(n-2) (?i-r+l). 

4.  16.  5.  120;  720. 

6.  6  ;  24  ;  720  ;  n{n-\)  (?i-2) 3-2  1. 

7.  19958400.  8.  50400. 

9.  35;190;l;^^V^);^^^^-^>-V^^^-^^+^^   1. 

10.  4950.    11.  25.    12.  66.    13.  103740.    14.  581400. 
19.  l-2a^+|i»«-t?a?9+/Ta?i2_^\a7i5+,t^a7i8. 


58  ANSWERS  [Ex.  for  Rev.  VII-VllI 

20.  l+^x+-g\x-2+-sUj^+ •     3*-  '^''h 

21.  „Cr-ia»-'+i6'"-i.         '  36.  4+8+12+16+20. 
22    — £c20  38.  ?^. 

23.  --'|Ja'2o.  39.  a-i;ri=— 4. 

24.  l+JJ.r-5.r3+3a^-£C«.  41.  f||Mf- 

25.  2.828  ;  2.924  ;  2.024.  42.  11|. 

28.  39.  43.   l+3i+6+8i+ll  +  J3i+16. 

29.  —V.  ^'  ^' 

30.  '^"^  ;  50.  45.  4i/3. 

31.  11.  .  46.  7 ±41/3. 

33.  345.  47.  3+6+18+54+162. 

Exercises  for  Review  (VIII). 
1.  3or-t.  2.    ±1/5.  3.-1,-1,2. 

4 a+V2b-a'^  a-y2h-^\  ^_a-V2h-a'^ 

7/=r«+l/26— a2 

^     (i;^=6 

5.  1,  -2,  3,  3.  6.  65780.  7.   (?i+l)th. 

11.  l+6a7+18x2+54.T3+162.T*+  •  •  •  ■  . 

12.  l+3a?+-4ic2+7ic3+lla?4+ 

13.  l-2a;+a?2+a?3-2a74+ 

14.  l-|ic-V-£c2— V/^-W/^^- • 

15.  a;=2/+37/2+13^3+672/*+ 

on     (f,\  1 I  "         1  .    /i)\  q 1 I  "  , 

•   ^^   2(1-.^) "^2(1+0;)     (1+07)2'.^'^'     2(a+l)'^2(rt-l)  ' 
(o\      3  2  1.   ,,'22  1 

^^  a;+l    2.'r+l"^2a;-l'  ^   ^  a;    a;+l'^(a?+l)2  ' 

(p\  ^-1 5    _      4      .  ,  -.   3     1_      3  5a?-3 

'"^  a?    a:2    a;+i     (a?+l)2' ^-^  ^  ir2    a;    2(a?-l)"^2(.T2+a;+i)* 

22.  7  ;  3  ;  f.  26.  10^=65. 

23.  1  and  2  ;  2  and  3  ;  3  and  4.  27.  2  ;  3  ;  -4  ;  2  ;  a? ;  0  ;  1. 
30.  (a)  logaOJ+loga?/;  (5)  logaa;-loga^ ;  (c)  nlogaO?;  (cZ)  i- logaO?. 

32.  (a)  1 ;  (5)  0 ;  (c)  -1  ;  (d)  -3  ;  (e)  1. 

33.  1.  34.  An  integer  plus  a  fraction. 


ANSWERS  TO  EXERCISES  IN  APPENDIX 


Exercise  I. 

1.  ^x+1. 

2.  x^-5y. 

6. 

7. 

1+x—x'^+x^. 
1x^+8x^-2. 

11.    «+62+|. 

3.  x^-x+2. 

8. 

x^-xy+y^. 

12.    X+4:--^. 

4.  2a2+5a-7 

9. 

x^-Sxy+2y^. 

X 

13.  .^-"-f  ^ 
2     2x 

5.  a2-2a-2. 

10. 

y^+^y+l- 

14.  l+4^x-ix^+ 

•   •  . 

16.  l4-ia2. 

-ia4+   •••••. 

15.  1-x—^x^- 

-   • 

•  • 

17.  m^+l,m-^-^^m-^+   •  •  •  •  . 

Exercise  II. 

1.  86. 

5. 

3.5. 

9.  .162. 

13.  .774  +  . 

2,  43. 

6. 

12.1. 

10.  51.1. 

14.  10.246+. 

3.  162. 

7. 

11.2. 

11.  2.236+ 

15.  19.261  +  . 

4.  203. 

,8. 

2.15. 

12.  1.732+ 
17.  3.435+. 

Exercise  III. 

16.  .866+. 

1.  x+2y. 

6. 

X-2-X+1. 

10.  ar2-2;r-l. 

2.  0^2-1.^ 

3.  a;2+32/2. 

7. 
8. 

l-3a+2a2. 

2+i!t;+£t'2. 

11.  a;2-2a;-3. 

4.  3a -462. 

5.  l-2a. 

9. 

a-4+2. 
a 

Exercise    IV. 

''■  M 

1.  47. 

4. 

53. 

7.  806.               10. 

3.45.              13.  4.73+ 

2.  14. 

5. 

83. 

8.  3.9.                11 

1.25+.          14.  1.54+, 

3.  25. 

6. 

698 

9.  6.9.                12. 
59 

2.15+.          15.  .53+. 

30 

ANSWERS 

[Ex. 

IN  App 

Exercise  V. 

1. 

a;-2. 

7. 

x+1. 

13.  x^y—xy. 

19. 

x^+x^- 

2. 

x-2. 

8. 

a-2. 

14.  1ab-ab^. 

20. 

a-1. 

3. 

x+1. 

9. 

m^-2m-S. 

15.  £c3+2a;2. 

21. 

m-1. 

4. 

2a-l. 

10. 

da+4b. 

16.  None. 

22. 

x-2. 

5. 

2x-9. 

11. 

2x^+xy. 

17.  x+1. 

23. 

a+1. 

6. 

a2+4a-5. 

12. 

a2_a6-2b2. 

18.  None. 

24. 

a -10. 

25.  y+2. 

Exercise  VI. 

6.  4n5-15n3-33w2+20n. 

7.  x^-2x^-Qx^+20x'^-ldx+e. 

8.  2a?4+7a^-2a;2-13.r+6. 

9.  12a4+44a8+21a2-47a-30. 
10.  x^+4x^-Sx^-29x^+2x+24. 

11.  2x^+2Sx^+Q0x^+28x-'S2. 

12.  18a8+84a7+162a6+186a5+132a4+54a8+12a2. 


1.  4a8-20a2+31a-15. 

2.  24a8+50a2_31a-70. 

3.  ;r6-17a?3+12£c2+52.T-48. 

4.  x^-'i7x^+lQx^+Q0x-80. 

5.  48a4+64a3+37a2+10a+l 


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